THEORY OF SUPERCONDUCTIVITYA PrimerbyHelmut Es hrigDire tor of the Institute for Solid State and Materials Resear h andProfessor for Solid State Physi s, Dresden University of Te hnology

These le ture notes are dedi ated, on the o assion ofhis eighties birthday on 4 July 2001,to Musik Kaganovwho has done me the honor and delight of his friendshipover de ades and whom I owe a great part of an attitudetowards Theoreti al Physi s.

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Contents1 INTRODUCTION 42 PHENOMENA, LONDON THEORY 52.1 Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 London theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Gauge symmetry, London gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 THE THERMODYNAMICS OF THE PHASE TRANSITION 133.1 The Free Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 The Free Enthalpy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 The thermodynami riti al �eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4 Heat apa ity jump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 THE GINSBURG-LANDAU THEORY; TYPES OF SUPERCONDUCTORS 174.1 The Landau theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.2 The Ginsburg-Landau equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.3 The Ginsburg-Landau parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.4 The phase boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.5 The energy of the phase boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 INTERMEDIATE STATE, MIXED STATE 255.1 The intermediate state of a type I super ondu tor . . . . . . . . . . . . . . . . . . . . 255.2 Mixed state of a type II super ondu tor . . . . . . . . . . . . . . . . . . . . . . . . . . 275.3 The ux line in a type II super ondu tor . . . . . . . . . . . . . . . . . . . . . . . . . . 286 JOSEPHSON EFFECTS 326.1 The d. . Josephson e�e t, quantum interferen e . . . . . . . . . . . . . . . . . . . . . . 336.2 The a. . Josephson e�e t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 MICROSCOPIC THEORY: THE FOCK SPACE 377.1 Slater determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.2 The Fo k spa e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387.3 O upation number representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397.4 Field operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 MICROSCOPIC THEORY: THE BCS MODEL 428.1 The normal Fermi liquid as a quasi-parti le gas . . . . . . . . . . . . . . . . . . . . . . 428.2 The Cooper problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448.3 The BCS Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468.4 The Bogoliubov-Valatin transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 479 MICROSCOPIC THEORY: PAIR STATES 519.1 The BCS ground state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519.2 The pair fun tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529.3 Non-zero temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5310 MICROSCOPIC THEORY: COHERENCE FACTORS 5510.1 The thermodynami state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5510.2 The harge and spin moment densities . . . . . . . . . . . . . . . . . . . . . . . . . . . 5510.3 Ultrasoni attenuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5610.4 The spin sus eptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573

1 INTRODUCTIONThese le ture notes introdu e into the phenomenologi al and qualitative theory of super ondu tiv-ity. Nowhere any spe i� assumption on the mi ros opi me hanism of super ondu tivity is madealthough on a few o asions ele tron-phonon intera tion is mentioned as an example. The theoreti alpresuppositions are ex lusively guided by phenomena and kept to a minimum in order to arrive atresults in a reasonably simple manner.At present there are indi ations of non-phonon me hanisms of super ondu tivity, yet there is nohard proof up to now. The whole of this treatise would apply to any me hanism, possibly withindi ated modi� ations, for instan e a symmetry of the order parameter di�erent from isotropy whi hhas been hosen for the sake of simpli ity.This is a primer. For ea h onsidered phenomenon, only the simplest ase is treated. Referen esare given basi ally to the most important seminal original papers. Despite the above mentionedstri t phenomenologi al approa h the te hni al presentation is standard throughout, so that it readily ompares to the existing literature.1More advan ed theoreti al tools as �eld quantization and the quasi-parti le on ept are introdu edto the needed level before they are used. Basi notions of Quantum Theory and of Thermodynami s(as well as of Statisti al Physi s in a few o asions) are presupposed as known.In Chapter 2, after a short enumeration of the essential phenomena of super ondu tivity, theLondon theory is derived from the sole assumption that the super urrent as an ele tri al urrent isa property of the quantum ground state. Thermoele tri s, ele trodynami s and gauge properties aredis ussed.With the help of simple thermodynami relations, the ondensation energy, the thermodynami riti al �eld and the spe i� heat are onsidered in Chapter 3.In Chapter 4, the Ginsburg-Landau theory is introdu ed for spatially inhomogeneous situations,leading to the lassi� ation of all super ondu tors into types I and II. The simplest phase diagram ofan isotropi type II super ondu tor is obtained in Chapter 5.The Josephson e�e ts are qualitatively onsidered on the basis of the Ginsburg-Landau theory inChapter 6. Both, d. . and a. . e�e ts are treated.The remaining four hapters are devoted to the simplest phenomenologi al weak oupling theoryof super ondu tivity on a mi ros opi level, the BCS theory whi h gave the �rst quantum theoreti alunderstanding of super ondu tivity, 46 years after the experimental dis overy of the phenomenon. Forthis purpose, in Chapter 7 the Fo k spa e and the on ept of �eld quantization is introdu ed. Then,in Chapter 8, the Cooper theorem and the BCS model are treated with o upation number operatorsof quasi-parti le states whi h latter are introdu ed as a working approximation in Solid State Physi s.The nature of the harged bosoni ondensate, phenomenologi ally introdu ed in Chapter 2, is derivedin Chapter 9 as the ondensate of Cooper pairs. The ex itation gap as a fun tion of temperature ishere the essential result. The treatise is losed with a onsideration of basi examples of the importantnotion of oheren e fa tors.By spe ifying more details as lower point symmetry, real stru ture features of the solid (for instan e ausing pinning of vortex lines) and many more, a lot of additional theoreti al onsiderations wouldbe possible without spe ifying the mi ros opi me hanism of the attra tive intera tion leading tosuper ondu tivity. However, these are just the notes of a one-term two-hours le ture to introdu einto the spirit of this kind of theoreti al approa h, not only addressing theorists. In our days of livelyspe ulations on possible auses of super ondu tivity it should provide the new ommer to the �eld(again not just the theorist) with a safe ground to start out.1Two lassi s are re ommended for more details: J. R. S hrie�er, Theory of Super ondu tivity, Benjamin, New York,1964, and R. D. Parks (ed.), Super ondu tivity, vol. I and II, Dekker, New York, 1969.4

Figure 1: Resistan e in ohms of a spe imen of mer ury versus absolute temperature. This plot byKamerlingh Onnes marked the dis overy of super ondu tivity. (Taken from: Ch. Kittel, Introdu tionto Solid State Physi s, Wiley, New York, 1986, Chap. 12.)2 PHENOMENA, LONDON THEORYHelium was �rst lique�ed by Kammerling Onnes at Leiden in 1908. By exhausting the helium vaporabove the liquid the temperature ould soon be lowered down to 1.5K.Shortly afterwards, in the year 1911, it was found in the same laboratory1 that in pure mer urythe ele tri al resistan e disappeared abruptly below a riti al temperature, T = 4.2K.Deliberately in reasing ele tron s attering by making the mer ury impure did not a�e t the phe-nomenon. Shortly thereafter, the same e�e t was found in indium (3.4K), tin (3.72K) and in lead(7.19K). In 1930, super ondu tivity was found in niobium (T = 9.2K) and in 1940 in the metalli ompound NbN (T = 17.3K), and this remained the highest T until the 50's, when super ondu tivityin the A15 ompounds was found and higher T -values appeared up to T = 23.2K in Nb3Ge, in 1973.These materials were all normal metals and more or less good ondu tors.In 1964, Marvin L. Cohen made theoreti al predi tions of T -values as high as 0.1K for ertaindoped semi ondu tors, and in the same year and the following years, super ondu tivity was found inGeTe, SnTe (T � 0.1K, ne � 1021 m�3) and in SrTiO3 (T = 0.38K at ne � 1021 m�3, T �0.1K at ne � 1018 m�3).In the early 80's, super ondu tivity was found in several ondu ting polymers as well as in \heavyfermion systems" like UBe13 (T � 1K in both ases). The year 2000 Nobel pri e in Chemistry wasdedi ated tho the predi tion and realization of ondu ting polymers (syntheti metals) in the late70's.1H. K. Onnes, Commun. Phys. Lab. Univ. Leiden, No124 (1911); H. K. Onnes, Akad. van Wetens happen(Amsterdam) 14, 818 (1911). 5

1900 1920 1940 1960 1980 2000

YEAR

20

40

60

80

100

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Tc(K

)

H2

He

[[

[

Pb

Hg Nb

NbC

NbN

V3Si

Nb3Sn

Nb−Al−Ge

Nb3Ge

(La−Ba)−214

April, 1986

December, 1986

(under pressure)

December, 1986(under pressure)

Y−123

Bi−2223

Tl−2223

Hg−1223

January, 1987

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April, 1993September, 1992February, 1988

September, 1993(under pressure)

August, 1993(under pressure)

LiquidCF

4

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LiquidN

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Liquid

Figure 2: The evolution of T with time (from C. W. Chu, Super ondu tivity Above 90K and Beyondin: B. Batlogg, C. W. Chu, W. K. Chu D. U. Gubser and K. A. M�uller (eds.) Pro . HTS Workshopon Physi s, Materials and Appli ations, World S ienti� , Singapore, 1996.).In 1986, Georg Bednorz and Alex M�uller found super ondu tivity in (La,Sr)2CuO4 with T = 36K,an in redible new re ord.1Within months, T -values in uprates were shooting up, and the re ord is now at T � 135K.2.1 PhenomenaZero resistan e2 No resistan e is dete table even for high s attering rates of ondu tion ele trons.Persistent urrents magneti ally indu ed in a oil of Nb0:75Zr0:25 and wat hed with NMR yielded anestimate of the de ay time greater than 105 years! (From theoreti al estimates the de ay time maybe as large as 101010 years!)Absen e of thermoele tri e�e ts3 No Seebe k voltage, no Peltier heat, no Thomson heat isdete table (see next se tion).Ideal diamagnetism �m = �1. Weak magneti �elds are ompletely s reened away from the bulkof a super ondu tor.Meissner e�e t4 If a super ondu tor is ooled down in the presen e of a weak magneti �eld, belowT the �eld is ompletely expelled from the bulk of the super ondu tor.Flux quantization5 The magneti ux through a super ondu ting ring is quantized and onstantin time. This phenomenon was theoreti ally predi ted by F. London in 1950 and experimentallyveri�ed 1961.1J. G. Bednorz and K. A. M�uller, Z. Phys. B64, 189 (1986).2J. File and R. G. Mills, Phys. Rev. Lett. 10, 93 (1963).3W. Meissner, Z. Ges. K�alteindustrie 34, 197 (1927).4W. Meissner and R. O hsenfeld, Naturwiss. 21, 787 (1933).5B. S. Deaver and W. M. Fairbank, Phys. Rev. Lett. 7, 43 (1961); R. Doll and M. N�abauer, Phys. Rev. Lett. 7,51 (1961). 6

2.2 London theory1Phenomena (a) and (b) learly indi ate that the super urrent (at T = 0) is a property of the quantumground state:There must be an ele tri ally harged ( harge quantum q), hen e omplex bosoni �eldwhi h ondenses in the ground state into a ma ros opi amplitude:nB = jj2; (1)where nB means the bosoni density, and is the orresponding �eld amplitude.Sin e the �eld is ele tri ally harged, it is subje t to ele tromagneti �elds (E;B) whi h are usuallydes ribed by potentials (U;A): E = ��A�t � �U�r ; (2a)B = ��r �A: (2b)The �eld amplitude should obey a S hr�odinger equation12mB �~i ��r � qA�2+ qU = �E � �B�; (3)where the energy is reasonably measured from the hemi al potential �B of the boson �eld, sin e whatis measured in a voltmeter is rather the ele tro hemi al potential� = �B + qU (4)than the external potential U , or the e�e tive ele tri �eldEe� = ��A�t � 1q ���r : (5)As usual in Quantum Me hani s, �i~�=�r is the anoni al momentum and (�i~�=�r� qA) = pm isthe me hani al momentum.The super urrent density is thenjs = q pmmB nB = qmB<��pm� = � iq~2mB �� ��r� ��r��� q2mB�A: (6)It onsists as usual of a `paramagneti urrent' (�rst term) and a `diamagneti urrent' (se ond term).2In a homogeneous super ondu tor, where nB = onst., we may write(r; t) = pnBei�(r;t); (7)and have �js = ~q ���r �A; � = mBnBq2 : (8)Sin e in the ground state E = �, and E = i~�=�t, we also have~���t = ��: (9)The London theory derives from (8) and (9). It is valid in the London limit, where nB = onst. inspa e an be assumed.1F. London and H. London, Pro . Roy. So . A149, 71 (1935); F. London, Pro . Roy. So . A152, 24 (1935); F.London, Super uids, Wiley, London, 1950.2These are formal names: sin e the splitting into the two urrent ontributions depends on the gauge, it has nodeeper physi al meaning. Physi ally, paramagneti means a positive response on an external magneti �eld (enhan ingthe �eld inside the material) and diamagneti means a negative response.7

The time derivative of (8) yields with (9)���js��t = ��A�t � 1q ���r ;or ���js��t = Ee� (10)This is the �rst London equation:A super urrent is freely a elerated by an applied voltage, or, in a bulk super ondu torwith no super urrent or with a stationary super urrent there is no e�e tive ele tri �eld( onstant ele tro hemi al potential).The �rst London equation yields the absen e of thermoele tri e�e ts, if the ele tro hemi al poten-tials of ondu tion ele trons, �el, and of the super urrent, �, are oupled. The thermoele tri e�e tsare sket hy illustrated in Fig. 3. The �rst London equation auses the ele tro hemi al potential ofthe super urrent arrying �eld to be onstant in every stationary situation. If the super urrent ar-rying �eld rea ts with the ondu tion ele tron �eld with n ele trons forming a �eld quantum with harge q, then the ele tro hemi al potentials must be related as n�el = �. Hen e the ele tro hemi alpotential of the ondu tion ele trons must also be onstant: no thermopower (Seebe k voltage) maydevelop in a super ondu tor. The thermoele tri urrent owing due to the temperature di�eren eis an eled by a ba k owing super urrent, with a ontinuous transformation of ondu tion ele tronsinto super urrent density at the one end of the sample and a ba k transformation at the other end.If a loop of two di�erent normal ondu tors is formed with the jun tions kept at di�erent tem-peratures, then a thermoele tri urrent develops together with a di�eren e of the ele tro hemi alpotentials of the two jun tions, and several forms of heat are produ ed, everything depending on the ombination of the two metals. If there is no temperature di�eren e at the beginning, but a urrent ismaintained in the ring (by inserting a power supply into one of the metal halfs), then a temperaturedi�eren e between the jun tions will develop. This is how a Peltier ooler works. In a loop of twosuper ondu tors non of those phenomena an appear sin e a di�eren e of ele tro hemi al potentials annot be maintained. Every normal urrent is lo ally short- ir uited by super urrents.If, however, a normal metal A is ombined with a super ondu tor B in a loop, a thermoele tri urrent will ow in the normal half without developing an ele tro hemi al potential di�eren e of thejun tions be ause of the presen e of the super ondu tor on the other side. This yields a dire t absolutemeasurement of the thermoele tri oeÆ ients of a single material A.The url of Eq. (8) yields (with ��r � ��r = 0)��r � ��js� = �B: (11)This is the se ond London equation. It yields the ideal diamagnetism, the Meissner e�e t, and the ux quantization.Take the url of Maxwell's equation (Ampere's law) and onsider ��r�� ��r�B� = ��r � �B�r �� �2�r2B:��r �B = �0�js + j�; �B�r = 0; (12)��r � � ��r �B� = �0 ��r � �js + j�;� �2�r2B = �0 ��r � �js + j�;�2�r2B = �0� B � �0 ��r � j:8

normal ondu torT1 < T2Seebe k voltage�2 � �1 � T2 � T1fT1(�� �1;el)

� super ondu torT1 < T2

ne! q q ! ne�1;el �2;el n�1;el = � super ondu torj � �Tsuper urrent �j n�2;el = �normal ondu tor thermoele tri urrentSin e j(t) = onst.) js(t) = onst.) Ee� = ��=q = 0; �� = 0due to the �rst London equation.fT2(�� �2;el)The di�eren e of the Fermi distributionfun tions fT in onne tion with anon- onstant density of states results ina di�eren e of ele tro hemi al potentials �due to the detailed balan e of urrents.

AB T2<T1

thermoele tri urrent J Ohm's heat +Thomson heat�A(�T; J)Peltier heat�AB(T2; J)Peltier heat�BA(T1; J)

loop of two metals A and B: no total urrentno heatAB T2<T1

JsJA

B T2<T1super ondu tornormal ondu torOhm's heat +Thomson heat�B(��T; J)

Figure 3: Thermoele tri phenomena in normal ondu tors and super ondu tors.9

��������

�LzBjs � e�z=�L xz yexternal �eld B0

super ondu torFigure 4: Penetration of an external magneti �eld into a super ondu tor.If j = 0 or ��r � j = 0 for the normal urrent inside the super ondu tor, then�2�r2B = B�2L ; �L =s ��0 =r mBnB�0q2 (13)with solutions B = B0e�n�r=�L ; n2 = 1; n �B0 = 0 (14)several of whi h with appropriate unit ve tors n may be superimposed to ful�ll boundary onditions.�L is London's penetration depth.Any external �eld B is s reened to zero inside a bulk super ondu ting state within a surfa e layerof thi kness �L. It is important that (11) does not ontain time derivatives of the �eld but the �eldB itself: If a metal in an applied �eld B0 is ooled down below T , the �eld is expelled.C

Bd� �Figure 5: Flux through a super ondu ting ring.

Consider a super ondu tingring with magneti ux � pass-ing through it (Fig. 5). Be ause of(14) and (12), js = 0 deep insidethe ring on the ontour C. Hen e,from (10), Ee� = E = 0 there.From Faraday's law, (�=�r)�E =�B,d�dt = ddt ZABdS = � IC Edl = 0;(15)where A is a surfa e with bound-ary C, and � is the magneti uxthrough A.Even if the super urrent in asurfa e layer of the ring is hang-ing with time (for instan e, if anapplied magneti �eld is hanging with time), the ux � is not:The ux through a super ondu ting ring is trapped.Integrate Eq. (8) along the ontour C:IC�A+�js� � dl = ~q IC ���r � dl:10

The integral on the right hand side is the total hange of the phase � of the wavefun tion (7) aroundthe ontour, whi h must be an integer multiple of 2� sin e the wavefun tion itself must be unique.Hen e, IC�A+�js� � dl = ~q 2�n: (16)The left hand integral has been named the uxoid by F. London. In the situation of our ring we �nd� = ~q 2�n: (17)By dire tly measuring the ux quantum �0 the absolute value of the super ondu ting harge wasmeasured: jqj = 2e; �0 = h2e: (18)(The sign of the ux quantum may be de�ned arbitrarily; e is the proton harge.)If the super urrent js along the ontour C is non-zero, then the ux � is not quantized any more,the uxoid (16), however, is always quantized.In order to determine the sign of q, onsider a super ondu ting sample whi h rotates with theangular velo ity !. Sin e the sample is neutral, its super ondu ting harge density qnB is neutralizedby the harge density �qnB of the remainder of the material. Ampere's law (in the absen e of anormal urrent density j inside the sample) yields now��r �B = �0�js � qnBv�;where v = ! � r is the lo al velo ity of the sample, and js is the super urrent with respe t to therest oordinates. Taking again the url and onsidering��r � v = ��r � �! � r� = !�r�r � �! � ��r�r = 3! �! = 2!leads to � �2�r2B = �0 ��r � js � 2�0qnB!:We de�ne the London �eld BL � �2�2L�0qnB! = �2mBq ! (19)and onsider the se ond London equation (11) to obtain�2�r2B = B �BL�2L : (20)Deep inside a rotating super ondu tor the magneti �eld is not zero but equal to the homogeneousLondon �eld.Independent measurements of the ux quantum and the London �eld result inq = �2e; mB = 2me: (21)The bosoni �eld is omposed of pairs of ele trons.11

2.3 Gauge symmetry, London gaugeIf �(r; t) is an arbitrary di�erentiable single-valued fun tion, then the ele tromagneti �eld (2) isinvariant under the gauge transformationA �! A+ ���r ;U �! U � ���t : (22a)Sin e potentials in ele trodynami s an only indire tly be measured through �elds, ele trodynami sis symmetri with respe t to gauge transformations (22a).Eqs. (8, 9), and hen e the London theory are ovariant under lo al gauge transformations, if (22a)is supplemented by � �! � � 2e~ �;� �! �� ���t : (22b)From (8), the super urrent js is still gauge invariant, and so are the ele tromagneti properties of asuper ondu tor. However, the ele tro hemi al potential � is dire tly observable in thermodynami sby making onta t to a bath. The thermodynami super ondu ting state breaks gauge symmetry.For theoreti al onsiderations a spe ial gauge is often advantageous. The London gauge hooses �in (22b) su h that the phase � � 0: Then, from (8),�js = �A; (23)whi h is onvenient for omputing patterns of super urrents and �elds.

12

3 THE THERMODYNAMICS OF THEPHASE TRANSITION1Up to here we onsidered super ondu tivity as a property of a bosoni ondensate. From experimentwe know, that the onsidered phenomena are present up to the riti al temperature, T , of the tran-sition from the super ondu ting state, indexed by s, into the normal ondu ting state, indexed byn, as temperature rises. The parameters of the theory, nB and �L, are to be expe ted temperaturedependent: nB must vanish at T .In this and the next hapters we onsider the vi inity of the phase transition, T � T � T .3.1 The Free EnergyExperiments are normally done at given temperature T , pressure p, and magneti �eld B. Sin ea ording to the �rst London equation (10) there is no stationary state at E 6= 0, we must keep E = 0in a thermodynami equilibrium state. Hen e, we onsider the (Helmholtz) Free EnergyFs(T; V;B); Fn(T; V;B); (24)�F�T = �S; �F�V = �p; �F�B = �Vm; (25)where S is the entropy, and m is the magnetization density. First, the dependen e of Fs on B isdetermined from the fa t that in the bulk of a super ondu torBext +Bm = B + �0m = 0 (26)as it follows from the se ond London equation (11). Hen e,�Fs�B = +VB�0 =) Fs(B) = Fs(0) + V B22�0 : (27)The magneti sus eptibility of a normal (non-magneti ) metal isj�m;nj � 1 = j�m;sj; (28)hen e it may be negle ted here: Fn(B) � Fn(0): (29)Eq. (27) implies ( f. (25)) Fs(T; V;B) = Fs(T; V; 0) + V B22�0 ;p(T; V;B) = p(T; V; 0)� B22�0 : (30)The pressure a super ondu tor exerts on its surroundings redu es in an external �eld B: The �eld Bimplies a for e per area F = �n B22�0 (31)on the surfa e of the super ondu tor with normal n.1L. D. Landau and E. M. Lifshits, Ele trodynami s of Continuous Media, Chap. VI, Pergamon, Oxford, 1960.13

3.2 The Free EnthalpyThe relations between the Free Energy F and the Free Enthalpy (Gibbs Free Energy) G at B = 0and B 6= 0 read Fs(T; V; 0) = Gs(T; p(T; V; 0); 0)� p(T; V; 0)Vand Fs(T; V; 0) + V B22�0 = Fs(T; V;B) == Gs(T; p(T; V;B);B)� p(T; V;B)V == Gs(T; p(T; V; 0)� B22�0 ;B)� p(T; V; 0)V + V B22�0 :These relations ombine toGs(T; p(T; V; 0); 0) = Gs(T; p(T; V; 0)� B22�0 ;B);or Gs(T; p;B) = Gs(T; p+ B22�0 ; 0): (32)In a ord with (31), the e�e t of an external magneti �eld B on the Free Enthalpy is a redu tion ofthe pressure exerted on the surroundings, by B2=2�0. In the normal state, from (29),Gn(T; p;B) = Gn(T; p; 0): (33)The riti al temperature T (p;B) is given byGs(T ; p+ B22�0 ; 0) = Gn(T ; p; 0): (34a)Likewise B (T; p) from Gs(T; p+ B2 2�0 ; 0) = Gn(T; p; 0): (34b)BT TT (B)B B (T )

Figure 6: The riti al temperature as a fun tion of the applied magneti �eld and the thermodynami riti al �eld as a fun tion of temperature. 14

3.3 The thermodynami riti al �eldThe Free Enthalpy di�eren e between the normal and super ondu ting states is usually small, so thatat T < T (B = 0) the thermodynami riti al �eld B (T ) for whi h (34) holds is also small. Taylorexpansion of the left hand side of (34) yieldsGn(T; p) = Gs(T; p) + B2 2�0 �Gs�p = Gs(T; p) + B2 2�0V (T; p;B = 0): (35)Experiment shows that at B = 0 the phase transition is se ond order,Gn(T; p)�Gs(T; p) = a�T (p)� T �2: (36)Hen e, B (T; p) = b�T (p)� T �; (37)where a is a onstant, and b =p2�0a=V . T (p) is meant for B = 0.B B (T ) P TT (p)

b(T (p)� T )�B �T = 0FIG. 7: The thermodynami riti al �eld.

�0M = V Bm�0�G�m = �1 Meissnere�e t

Bnormal stateB (T )

FIG. 8: The magnetization urve of a super ondu tor.We onsider all thermodynami parameters T; p;B at the phase transition point P of Fig. 7. From(32), Ss(T; p;B) = ��Gs�T = Ss(T; p+ B22�0 ; 0);Vs(T; p;B) = �Gs�p = Vs(T; p+ B22�0 ; 0): (38)Di�erentiating (34b) with respe t to T yields, with (38),��T Gs(T; p+ B2 (T; p)2�0 ; 0) = ��T Gn(T; p; 0 or B );�Ss(T; p;B ) + Vs(T; p;B )2�0 ��T B2 (T; p) = �Sn(T; p;B );�S(T; p;B ) = Ss(T; p;B )� Sn(T; p;B ) = Vs(T; p;B )�0 B (T; p)�B (T; p)�T : (39)15

A ording to (37) this di�eren e is non-zero for B 6= 0 (T < T (p)): For B 6= 0 the phase transitionis �rst order with a latent heat Q = T�S(T; p;B ): (40)For T ! 0, Nernst's theorem demands Ss = Sn = 0, and hen elimT!0 �B (T; p)�T = 0: (41)3.4 Heat apa ity jumpFor B � 0, T � T (p) we an use (35). Applying �T�2=�T 2 yields�Cp = Cp;s � Cp;n = �T �2�T 2�Gs(T; p)�Gn(T; p)� = TV (T; p)2�0 �2�T 2B2 (T; p): (42)The thermal expansion �V=�T gives a small ontribution whi h has been negle ted. With�2�T 2B2 = ��T 2B �B �T = 2 �B �T !2 + 2B �2B �T 2we �nd �Cp = TV�0 " �B �T !2 +B �2B �T 2 # (43)For T ! T (p), B ! 0 the jump in the spe i� heat is�Cp = T V�0 �B �T !2 = T V�0 b2: (44)It is given by the slope of B (T ) at T (p).STT (p)

B = 0Sn Ss

FIG. 9: The entropy of a super ondu tor.

Cp B = 0 �CpT (p) TFIG. 10: The heat apa ity of a super ondu tor.

16

4 THE GINSBURG-LANDAU THEORY;1TYPES OF SUPERCONDUCTORSA ording to the Landau theory of se ond order phase transitions with symmetry redu tion2 there is athermodynami quantity, alled an order parameter, whi h is zero in the symmetri (high temperature)phase, and be omes ontinuously non-zero in the less symmetri phase.4.1 The Landau theoryThe quantity whi h be omes non-zero in the super ondu ting state isnB = jj2: (45)For nB > 0, the ele tro hemi al potential � has a ertain value whi h breaks the global gauge sym-metry by �xing the time-derivative of the phase � of ( f. (22b)). A ording to the Landau theory,the Free Energy is the minimum of a \Free Energy fun tion" of the order parameter with respe t tovariations of the latter: F (T; V ) = min F(T; V; jj2): (46)

�������

���

F

jj2nB;min

T > T (t > 0)T = T (t = 0) T < T (t < 0)

Figure 11: The Free Energy fun tion.Close to the transition, for t = T � T T ; jtj � 1; (47)the order parameter jj2 is small, and F may be Taylor expanded (for �xed V ):F(t; jj2) = Fn(t) +A(t)jj2 + 12B(t)jj4 + � � � (48)From the �gure we see that A(t) T 0 for t T 0; B(t) > 0:1V. L. Ginsburg and L. D. Landau, Zh. Eksp. Teor. Fiz. (Russ.) 20, 1064 (1950).2L. D. Landau, Zh. Eksp. Teor. Fiz. (Russ.) 7, 627 (1937).17

Sin e jtj � 1, we put A(t) � �tV; B(t) � �V: (49)Then we have Fn(t) = Fn(t) for t � 0; (50)and 1V �F�jj2 = �t+ �jj2 = 0; that is,jj2 = ��t� ; Fs(t) = Fn(t)� �2t22� V for t < 0: (51)Re alling that small hanges in the Free Energy and Free Enthalpy are equal and omparing to(35) yields �2t22� = B2 2�0 =) B (t) = �jtjr�0� : (52)From (43), �Cp = T V�0 �B T �t!2 = VT �2� (53)follows. While �Cp an be measured, this is not always the ase for the thermodynami riti al �eld,B , as we will later see.Eqs. (51) and (52) may be rewritten asnB(t) = �� jtj; B2 (t) = �2t2�0� ;hen e, � = B2 (t)�0n2B(t) ; � = B2 (t)�0jtjnB(t) : (54)Sin e a ording to (37) B � t, it follows nB � t: (55)The bosoni density tends to zero linearly in T � T .4.2 The Ginsburg-Landau equationsIf we want to in orporate a magneti �eld B into the \Free Energy fun tion" (48), we have to realizethat B auses super urrents js � �=�r, and these reate an internal �eld, whi h was alled Bm in(26). The energy ontribution of must be related to (3). Ginsburg and Landau wrote it in the formF(t;B;) = Fn(t) ++ Z 1 d3rB2m2�0 + ZV d3r( ~24m ����� ��r + 2ie~ A�����2 + �tjj2 + �2 jj4); (56a)where also (21) was onsidered. The �rst orre tion term is the �eld energy of the �eld Bm reatedby , in luding the stray �eld outside of the volume V while 6= 0 inside V only. A is the ve torpotential of the total �eld a ting on : ��r �A = B +Bm: (56b)18

The Free Energy is obtained by minimizing (56a) with respe t to (r) and �(r). To prepare for avariation of �, the se ond integral in (56a) is integrated by parts:ZV d3r"� ��r + 2ie~ A�#"� ��r � 2ie~ A��# == � ZV d3r�� ��r + 2ie~ A�2+ Z�V d2n�� ��r + 2ie~ A�: (56 )From the �rst integral on the right we see that (56a) indeed orresponds to (3). The preferen e of thewriting in (56a) derives from that kineti energy expression being manifestly positive de�nite in anypartial volume.Now, the variation � ! � + Æ� yields0 != ÆF = ZV d3rÆ�(� ~24m� ��r + 2ie~ A�2 + �t+ �jj2) ++ Z�V d2nÆ� ~24m� ��r + 2ie~ A�:F is stationary for any variation Æ�(r), if14m�~i ��r + 2eA�2� �jtj+ �jj2 = 0 (57)and n�~i ��r + 2eA� = 0: (58)The onne tion of with Bm must be that of Ampere's law: (�=�r)�Bm = �0js with js givenby (6). Sin e in thermodynami equilibrium there are no urrents besides js in the super ondu tor,(�=�r)�B = 0 there. Hen e, we also have��r �Btot = �0js; Btot = B +Bm = ��r �A;js = ie~2m�� ��r� ��r��� 2e2m �A: (59)It is interesting to see that (59) is also obtained from (56a), if �, and A are varied independently:The variation of A on the left hand side of (56 ) yields2ie~ ZV d3rÆA � �� ��r � 2ie~ A�� ��� ��r + 2ie~ A��:With ÆBm = (�=�r)� ÆA the variation of the �rst integral of (56a) yieldsÆ Z 1 d3rB2m = 2 Z 1 d3rÆBm �Bm = 2 Z 1 d3r� ��r � ÆA� �Bm == 2 Z 1 d3r z }| {��r ��ÆA�Bm� = 2 Z 1 d3rÆA � � ��r �Bm� == 2 ZV d3rÆA � � ��r �Btot�+ � � � : (60)In the fourth equality an integration per parts was performed, and a � (b� ) = �b � (a� ) was used.(The over bra e indi ates the range of the di�erential operator.) Finally, the integral over the in�nite19

spa e is split into an integral over the super ondu tor (volume V ), where (�=�r)�Bm = (�=�r)�Btot,and the integral over the volume outside of the super ondu tor, indi ated by dots, sin e we do notneed it. Now, after adding the prefa tors from (56a) we see that stationarity of (56a) with respe t toa variation ÆA inside the volume V again leeds to (59).This situation is no a ident. From a more general point of view the Ginsburg-Landau fun tional(56a) may be onsidered as an e�e tive Hamiltonian for the u tuations of the �elds and A nearthe phase transition.1 This is pre isely the meaning of relating (56a) to (3).Eqs. (57) and (59) form the omplete system of the Ginsburg-Landau equations.The boundary ondition (58) omes about by the spe ial writing of (56a) without additional surfa eterms. This is orre t for a boundary super ondu tor/va uum or super ondu tor/semi ondu tor. A areful analysis on a mi ros opi theory level yields the more general boundary onditionn � �~i ��r + 2eA� = ib ; (61)where b depends on the outside material: b =1 for va uum or a non-metal, b = 0 for a ferromagnet,b �nite and non-zero for a normal metal.2In all ases, multiplying (61) by � and taking the real part yieldsn � js = 0 (62)as it must: there is no super urrent passing through the surfa e of a super ondu tor into the non-super ondu ting volume.Btot must be ontinuous on the boundary be ause, a ording to �Btot=�r = 0 and (59), itsderivatives are all �nite.4.3 The Ginsburg-Landau parameterTaking the url of (59) yields, like in (13),�2Btot�r2 = Btot�2 ; �2 = m2�0e2jj2 = m�2�0e2�jtj ; (63)where (51) was taken into a ount in the last expression. � is the Ginsburg-Landau penetration depth;it diverges at T like � � jtj�1: if T is approa hed from below, the external �eld penetrates more andmore, and eventually, at T , the diamagnetism vanishes.Eq. (57) ontains a se ond length parameter: In the absen e of an external �eld, A = 0, and forsmall , jj2 � �jtj=�, one is left with�2�r2 = �2 ; �2 = ~24m�jtj : (64)This equation des ribes spatial modulations of the order parameter jj2 lose to T . � is the Ginsburg-Landau oheren e length of su h order parameter u tuations. It has the same temperature dependen eas �, and their ratio, � = �� =s 2m2�~2�0e2 ; (65)is the elebrated Ginsburg-Landau parameter.1L. D. Landau and E. M. Lifshits, Statisti al Physi s, Part I, x147, Pergamon, London, 1980.2P. G. De Gennes, Super ondu tivity in metals and alloys, New York 1966, p. 225 �.20

Introdu tion of dimensionless quantitiesx = r=�; = ,s�jtj� ;b = B.p2B (t) = B���jtjr2�0� �;is = js���0.p2B (t)�;a = A.�p2�B (t)� (66)yields the dimensionless Ginsburg-Landau equations� 1i� ��x + a�2 � + j j2 = 0;��x � b = is; is = i2�� � ��x � ��x ��� �a (67)whi h ontain the only parameter �.4.4 The phase boundaryWe onsider a homogeneous super ondu tor at T . T in an homogeneous external �eld B � B (T ) inz-dire tion. We assume a plane phase boundary in the y�z-plane so that for x! �1 the material isstill super ondu ting, and the magneti �eld is expelled, but for x!1 the material is in the normalstate with the �eld penetrating.

issuper ondu ting normal

zy

bphaseboundary

b = 0 = 1 b = 1=p2 = 0x

Figure 12: Geometry of a plane phase boundary.We put = (x); bz = b(x); bx = by = 0;21

ay = a(x); ax = az = 0; b(x) = a0(x):Then, the super urrent is ows in the y-dire tion, and hen e the phase of depends on y. We onsider y = 0 and may then hoose real. Further, by �xing another gauge onstant, we may hoosea(0) = 0:Then, Eqs. (67) redu e to� 1�2 00 + a2 � + 3 = 0; a00 = a 2: (68)Let us �rst onsider �� 1: For large enough negative x we have a � 0 and � 1. We put = 1��(x),and get from the �rst equation (68)�00 � �2�1� �� 1 + 3�� = 2�2�; � � ep2�x; x . ��1:On the other hand, for large enough positive x we have b = 1=p2; a = xp2; � 1; hen e, again fromthe �rst equation (68), 00 � �2x22 ; � e��x2=2p2; �x2 � 1:The se ond Eq. (68) yields a penetration depth � �10 ; where 0 denotes the value of (x) where the�eld drops:1 b� e��x2=2p2

�� 11=� = � 1=p� > 1� e 0x 0 � p� 1� ep2�x 1p2

Figure 13: The phase boundary of a type I super ondu tor.In the opposite ase � � 1; falls o� for x & 1; where b � 1=p2; a � x=p2; and for x � 1; 00 � �2x2 =2 : 22

1 b �� 1� e��x2=2p2

1 = � 1=p� < 11p2

Figure 14: The phase boundary of a type II super ondu tor.4.5 The energy of the phase boundaryFor B = B (T ); b = 1 in our units, the Free Energy of the normal phase is just equal to the FreeEnergy of the super ondu ting phase in whi h b = 0; = 1: If we integrate the Free Energy densityvariation (per unit area of the y � z-plane), we obtain the energy of the phase boundary per area:�s=n = Z 1�1 dx( (B �B )22�0 + ~24m�j0j2 + 4e2~2 A2jj2�� �jtjjj2 + �2 jj4): (69)Sin e the external �eld is B , we have used Bm = Btot � Bext = B � B : In our dimensionlessquantities this is (x is now measured in units of �)�s=n = �B2 �0 Z 1�1 dx��b� 1p2�2 + 1�2 02 + �a2 � 1� 2 + 42 � == �B2 �0 Z 1�1 dx��a0 � 1p2�2 � 1�2 00 + �a2 � 1� 2 + 42 � == �B2 �0 Z 1�1 dx��a0 � 1p2�2 � 42 �: (70)First an integration per parts of 02 was performed, and then (68) was inserted. We see that �s=n anhave both signs: �s=n ? 0 for �a0 � 1p2�2 ? 42 or 2p2 7 � 1p2 � a0�:Sin e b must de rease if 2 in reases and = 0 at b = 1=p2, (b� 1=p2) = (a0 � 1=p2) and 2 musthave opposite signs whi h leads to the last ondition. If1p2 � a0 = 2p2would be a solution of (68), it would orrespond to �s=n = 0:23

We now show that this is indeed the ase for �2 = 1=2: First we �nd a �rst integral of (68): 00 = �2��a2 � 1� + 3�;2 0 00 = �2�2 0a2 � 2 0 + 2 3 0� == �2�2 0a2 + 2 2aa0 � 2a0a00| {z }�2 0 + 2 3 0�= 0 by the se ond Eq. (67) 02 = �2� 2a2 � a02 � 2 + 42 + onst.� (71)sin e 0 = = 0 for a0 = 1p2 =) onst. = 12 :Now we use �2 = 12 ; 1p2 � a0 = 2p2 ) �a00 = p2 0 = �a 2 ) 0 = �a p2and have from (71) 02 = 12�2 02 � a02 � �1�p2a0�+ � 1p2 � a0�2 + 12�;whi h is indeed an identity.Sin e 02=�2 > 0 enters the integral for �s=n in the �rst line of (70), it is lear that �s=n is positivefor �2 ! 0: Therefore, the �nal result is�s=n ? 0 for � 7 1p2 : type III (72)The names \type I" and \type II" for super ondu tors were oined by Abrikosov,1 and it was theexisten e of type II super ondu tors and a theoreti al predi tion by Abrikosov, whi h paved the wayfor te hni al appli ations of super ondu tivity.

1A. A. Abrikosov, Sov. Phys.{JETP 5, 1174 (1957). 24

5 INTERMEDIATE STATE, MIXED STATEIn Chapter 2 we onsidered a super ondu tor in a suÆ iently weak magneti �eld, B < B ; where wefound the ideal diamagnetism, the Meissner e�e t, and the ux quantization.In Chapter 3 we found that the di�eren e between the thermodynami potentials in the normaland the super ondu ting homogeneous phases per volume without magneti �elds may be expressedas ( f. (35)) 1V hGn(p; T )�Gs(p; T )i = 1V hFn(V; T )� Fs(V; T )i = B2 (T )2�0 (73)by a thermodynami riti al �eld B (T ): (We negle t here again the e�e ts of pressure or of orre-sponding volume hanges on B :)If a magneti �eld B is applied to some volume part of a super ondu tor, it may be expelled(Meissner e�e t) by reating an internal �eld Bm = �B through super urrents, on the ost of anadditional energy R d3rB2m=2�0 for the super ondu ting phase ( f. (56a)) and of a kineti energydensity (~2=4m)j(�=�r + 2ieA=~)j2 in the surfa e where the super urrents ow. If Bm > B ;the Free Energy of the super ondu ting state be omes larger than that of the normal state in ahomogeneous situation. However, B itself may ontain a part reated by urrents in another volumeof the super ondu tor, and phase boundary energies must also be onsidered. There are thereforelong range intera tions like in ferroele tri s and in ferromagnets, and orresponding domain patterns orrespond to thermodynami stable states. The external �eldB at whi h the phase transition appearsdepends on the geometry and on the phase boundary energy.5.1 The intermediate state of a type I super ondu torApply a homogeneous external �eld Bext to a super ondu tor. B = Bext+Bm depends on the shapeof the super ondu tor. There is a ertain point, at whi h B = Bmax > Bext (Fig. 15). If Bmax > B ;the super ondu ting state be omes instable there. On ould think of a normal-state on ave islandforming (Fig. 16).SCBmax

BextBext +BmFIG. 15: Total (external plus indu ed)magneti �eld around a type I super- ondu tor.

SCBmaxn

FIG. 16.25

This, however, annot be stable either: the point of Bmax = B has now moved into the super- ondu tor to a point of the phase boundary between the normal and super ondu ting phases, whi hmeans that in the shaded normal area B < B ; this area must be ome super ondu ting again (Fig. 16).Forming of a onvex island would ause the same problem (Fig. 17).n SCBmaxFIG. 17.

SCFIG. 18.

Bm BextB 23B sphere long rod�0�fenergy of the stray �eld ++ energy of phase boundariesg

�rst ordertransitionFigure 19:

What really forms isa ompli ated lamellous or�lamentous stru ture of al-ternating super ondu tingand normal phases throughwhi h the �eld penetrates(Fig. 18).The true magnetization urve of a type I super on-du tor in di�erent geome-tries is shown on Fig. 19.It depends on the geometrybe ause the �eld reated bythe shielding super urrentsdoes. In Se tion 3.C, forB 6= 0 the phase transitionwas obtained to be �rst or-der. Generally, the move-ment of phase boundaries ishindered by defe ts, hen ethere is hysteresis aroundB .26

5.2 Mixed state of a type II super ondu tor1If the phase boundary energy is negative, germs of normal phase may form well below B ; at the lower riti al �eld B 1; and germs of super ondu ting phase may form well above B ; at the upper riti al�eld B 2; in both ases by gaining phase boundary energy.At B / B 2; in small germs � 1 and b � onst. The dimensionless Ginsburg-Landau equations(67) may be linearized: � 1i� ��x + a�2 = : (74)yab

x .Figure 20:

We apply the external B-�eld in z-dire tion (Fig. 20),bx = by = 0; bz = b; ax = �by; ay = az = 0;and assume germ �laments along the �eld lines: = (x; y):Then, (74) is ast into"� 1i� ��x � by�2 � 1�2 �2�y2# = :With (x; y) = eipx�(y); p=(b�) = y0 this equation simpli�esto [�(1=�2)(d2=dy2)+b(y�y0)2℄� = �; or, after multiplyingwith �=2 and by de�ning u2 � �(y � y0)2 :"�12 d2du2 + 12b2u2#� = �2�: (75)This is the S hr�odinger equation for the ground state of a harmoni os illator with! = b = � =) B = p2�B :Hen e, B 2(T ) = p2�B (T ): (76)If a germ lose to the surfa e of a super ondu tor at y= onst. is onsidered, then the u- oordinatemust be ut at some �nite value. There, the boundary ondition (58) yields d�=du = 0 (sin e n�a = 0).Therefore, instead of the boundary problem, the symmetri ground state in a double os illator witha mirror plane may be onsidered (Fig. 21). 0(0) = 0 u0 uFIG. 21: The ground state of a double os illator. minu0 E0 = 0:59~!21A. A. Abrikosov, unpublished 1955; Sov. Phys.{JETP 5, 1174 (1957).27

Hen e, at B 3 = B 20:59 = 1:7B 2 = 2:4�B (77)super ondu tivity may set in in a surfa e layer of thi kness � �.In a type I super ondu tor, B 2 < B . However, only below B 2 germs with arbitrarily small -values ould form where the super ondu ting phase is already absolutely stable in a type I super- ondu tor. Here, for B 2 < B < B germs an only form with a non-zero minimal -value, whi h isinhibited by positive surfa e energy. In this region the super ooled normal phase is metastable. For� > 0:59=p2; B 3 > B ; and surfa e super ondu tivity may exist above B :5.3 The ux line in a type II super ondu torTo determine B 1; the opposite situation is onsidered. In the homogeneous super ondu ting statewith = 1; the external magneti �eld is tuned up until the �rst normal state germ is forming. Againwe may suppose that the germ is forming along a �eld line in z-dire tion. Sin e the phase boundaryenergy is negative, there must be a tenden y to form many fa e boundaries. However, the normalgerms annot form arbitrarily small sin e we know from (17) that the ux onne ted with a normalgerm in a super ondu tor is quantized and annot be smaller than�0 = 2�~2e = 2�p2B �� = 2�p2B �2� : (78)We onsider a ux line of total ux �0 along the z-dire tion (Fig. 22):Bz� � �jjFIG. 22: An isolated ux line. = jjei�; B = ezB(�); �2 = x2 + y2

One ould try to solve the Ginsburg-Landau equations for that ase. However, there is no generalanalyti solution, and the equations are valid lose to T only, where jj is small. Instead we assume� � 1; that is, � � �; and onsider only the region � � �; where jj = onst. Then, from Ampere'slaw and (6), ��r �B = �0js = �0e~m jj2 ���r � 2�0e2m jj2A == 2�0e2jj2m � ~2e ���r �A� = 1�2��02� ���r �A�;hen e, A+ �2 ��r �B = �02� ���r :We integrate this equation along a ir le around the ux line with radius � � � and use Stokes'theorem, H dsA = R d2n � ( ��r �A) :Z d2n �B + �2 I ds � � ��r �B� = �0: (79)28

The phase � must in rease by 2� on a ir le around one uxoid.Now, we take � � �� �: Then, the �rst term in (79) may be negle ted. We �nd�2���2 dBd� = �0 (80)or B(�) = �02��2 ln��0� �; �0 � �: (81)The integration onstant �0 was hosen su h that (81) vanishes for � & �; where a more a urateanalysis of (79) is ne essary to get the orre t asymptoti s.By applying Stokes' theorem also to the se ond integral of (79), we have for all �� �Z d2n � �B + �2 ��r �� ��r �B�� = Z d2n � �B � �2 �2�r2B� = �0:Sin e the right hand side does not hange if we vary the area of integration,B � �2 �2�r2B = 0must hold. In ylindri oordinates,�2�r2 = 1� ���� ��� + 1�2 �2��2 + �2�z2 ;hen e, B + �2 1� ���� ���B = 0or B00 + 1�B0 � 1�2B = 0: (82)This equation is of the Bessel type. Its for large � de aying solution isB(�) = �02��2K0� ��� ���!�!1 �0p8���2 e��=�; (83)where K0 is M Donald's fun tion (Hankel's fun tion with imaginary argument), and the oeÆ ienthas been hosen to meet (80) for �� �:The energy per length � of the ux line onsists of �eld energy and kineti energy of the super ur-rent: � = Z d2r� B22�0 + mB2 nBv2�; js = �2enBv; nB = jj2;� = Z d2r� b22�0 + m4e2nB j2s� == Z d2r� B22�0 + m4�0e2jj2� ��r �B�2� == Z d2r� B22�0 + �22�0� ��r �B� �� ��r �B�� == 12�0 Z d2rB � �B + �2 ��r �� ��r �B���� �22�0 I��� � I�!1!ds � �B �� ��r �B��:29

The �rst integrand was shown to vanish for �� �; and the last ontour integral vanishes for �!1.We negle t the ontribution from � . �; and �nd with (80))� � � �22�0 2��BdBd� � �0B(�)2�0 � �0B(0)2�0 : (84)With (81), the �nal result with logarithmi a ura y (ln(�=�)� 1) is� � �204��0�2 ln���� = �204��0�2 ln�: (85)This result also proves that the total energy is minimum for ux lines ontaining one uxoid ea h:For a ux line ontaining n uxoids the energy would be n2� while for n ux lines it would only ben� (� > 0).In this analysis, B was the �eld reated by the super urrent around the vertex line. Its intera tionenergy per length with a homogeneous external �eld in the same dire tion is� Z d2rBBext�0 = ��0Bext�0 : (86)(85) and (86) are equal at the lower riti al �eld Bext = B 1:B 1 = �04��2 ln � = B ln�p2�; �� 1: (87)The phase diagram of a type II super ondu tor is shown in Fig. 23. (There might be another phase withspatially modulated order parameter under ertain onditions, theoreti ally predi ted independentlyby Fulde and Ferell and by Larkin and Ovshinnikov; this FFLO phase has not yet been learly observedexperimentally.) BFFLO? surfa e

s s B 1B

B 2 B 3 nmixedphaseT TFigure 23: The phase diagram of a type II super ondu tor.A more detailed numeri al al ulation shows that in an isotropi material the energy is minimumfor a regular triangular latti e of the ux lines in the plane perpendi ular to them. From (87), atB = B 1 the density of ux lines is ln�=4��2; that is, the latti e onstant a1 is obtained froma21p3=2 = 4��2= ln�: a1 =s 8�p3 ln� � & �: (88)30

The lines (of thi kness �) indeed form nearly individually (Fig. 24). Sin e B 2 = B 12�2= ln�; thelatti e onstant a2 at B 2 is a2 = a1r ln�2�2 =s 4�p3 �� =s 4�p3 � & �: (89)The ores of the ux lines (of thi kness �) tou h ea h other while the �eld is already quite homoge-neous (Fig. 25). Sin e jj � 1 in the ore, the Ginsburg-Landau equations apply, and jj may rise ontinuously from zero: the phase transition at B 2 is se ond order.a1 & �B jj�

FIG. 24: Mixed phase for Bext � B 1.solutionof (4.3)jj

Ba2 & �FIG. 25: Mixed phase for Bext � B 2.For a long ylindri rod the stray �eld reated by super urrents outside of the rod may be negle ted,and one may express the �eld inside the rod asB = Bext � �0Mby a magnetization densityM: The hange in Free Energy at �xed T and V by tuning up the externalmagneti �eld is dF =MdBextFn � Fs = Z B 20 dBextM = B2 2�0 :

������������������������������������������������������

������������������������������������������������������

��������������������������������

���������������������

���������������������

�0M Bextjump orin�nite slope

B 2B B 1

Figure 26: Magnetization urve of a type II super ondu tor.The di�erently dashed areas in Fig. 26 are equal. For type II super ondu tors, B is only a theoreti alquantity. 31

6 JOSEPHSON EFFECTS1The quantitative des ription of Josephson e�e ts at T � T (the usual ase in appli ations) needs ami ros opi treatment. However, qualitatively they are the same at all temperatures T < T , hen equalitatively they may be treated within the Ginsburg-Landau theory.Consider a very thin weak link between two halfs of a super ondu tor (Fig. 27).2�S1 S2dx1 x21 2�2 = �1

x�2 = �1 + �FIG. 27: A weak link between two halfs of a super ondu tor.

(x1) � 1 = jjei�1(x2) � 2 = jjei�2

The order parameter has its thermodynami value on both sides x < x1; x > x2; but is expo-nentially small at x = 0: Hen e, any super urrent through the weak link is small, and may be onsidered onstant in both bulks of super ondu tor. In the weak link, not only jj is small, also itsphase may hange rapidly (e.g. from �2 = �1 to �2 = �1 + � by a very small perturbation).Without the right half, the boundary ondition (58) would hold at x1 :� ��x + 2ie~ Ax�����x1 = 0:In the presen e of the right half, this ondition must be modi�ed to slightly depending on the value2 : � ��x + 2ie~ Ax�����x1 = 2; (90)1B. D. Josephson, Phys. Lett. 1, 251 (1962). 32

where is a small number depending on the properties of the weak link. Time inversion symmetrydemands that (90) remains valid for ! �; A! �A; hen e must be real as long as the phase of does not depend on A. For the moment we hoose a gauge in whi h Ax = 0: Then, the super urrentdensity at x1 is js;x(x1) = ie~2m"�1 ��x ����x1 �1���x ����x1# == ie~2m h�12 �1�2i == jm2 �ei(�2��1) � ei(�1��2)� = jm sin��2 � �1�: (91)We generalize the argument of the sine fun tion by a general gauge transformation (22):� �! � � 2e~ �;A �! A+ ���r ; �Ax = 0! Ax = ����x�;� �! �� ���t�2 � �1 �! = �2 � �1 � 2e~ ��2 � �1� = �2 � �1 + 2e~ Z 21 dxAx; (92)d dt = 2e~ ��2 � �1�: (93)Re all that � is the ele tro hemi al potential measured by a voltmeter.Now, the general Josephson equation readsjs = jm sin : (94)6.1 The d. . Josephson e�e t, quantum interferen eA ording to (94), a d. . super urrent of any value between �jm and jm may ow through the jun tion,and a ording to (93) the potential di�eren e �2 � �1 is zero in that ase.Now, onsider a jun tion in the y� z-plane with a magneti �eld applied in z-dire tion. There aresuper urrents s reening the �eld away from the bulks of the super ondu ting halfs.������������

������������S1 S2

yxd � 2�

B 04 3b 1 2 (y)Figure 28: A Josephson jun tion in an external magneti �eld B.33

Let us onsider (y); and let (0) = 0 at the edge y = 0: We haveByd = Z1234 ds �A = Z 21 dxAx + Z 32 dyAy + Z 43 dxAx + Z 14 dyAy:In the jun tion we hoose the gauge Ax = �By; Ay = 0: Then, the y-integrals vanish, and from (92), (0) = �2 � �1 = 0; (y) = 0 + 2��0 Z 34 dxBy = 0 + 2�Bd�0 y:The d. . Josephson urrent density through the jun tion os illates with y a ording tojs(y) = jm sin 0 + 2�Bd�0 y!:In experiment, at B = 0 one always starts from a biased situation with js = jm; hen e 0 = �=2; andjs(y) = jm os 2�Bd�0 y: (95)The total urrent through the jun tion isIs = jm Z b0 dy os 2�Bd�0 y;where is the thi kness in z-dire tion. F = b is the area of the jun tion. WithZ b0 dy os(�y) = < Z b0 dyei�y = <ei�b � 1i� = sin(�b)�we �nd that, depending on the phase 0; the maximal urrent at a given �eld B isIs;max = Fjm j sin(2�Bbd=�0)j2�Bbd=�0 : (96)An even simpler situation appears, if one splits the jun tion into a double jun tion: Now, H ds�A =� is the magneti ux through the ut-out, and a � b = 2��0 I ds �A = 2� ��0 :

������������������

������������������ S2S1 B s a b

Figure 29: A simple SQUID geometry.34

a 2��=�0Is;max=2FjmFigure 30: Phase relations in Eq.(97).Hen e, the maximal d. . Josephson urrent isIs;max = Fjmmax a "sin a + sin� a + 2���0 �#:Is;max = 2Fjm����� os�� ��0������: (97)This is the basis to experimentally ount uxquanta with a devi e alled super ondu tingquantum interferometer (SQUID).6.2 The a. . Josephson e�e tWe return to (93) and (94), and apply a voltage �2 � �1 = V to the jun tion, that is, = (2e=~)V t:An a. . Josephson urrent js(t) = jm sin!Jt; !J = 2eV=~ (98)results, although a onstant voltage is applied. For a voltage of 10 �V a frequen y !J=2� = 4.8 GHzis obtained: the a. . Josephson e�e t is in the mi rowave region.If one overlays a radio frequen y voltage over the onstant voltage,�2 � �2 = V + Vr os(!rt); (99)one obtains a frequen y modulation of the a. . Josephson urrent:js = jm sin�!Jt+ 2eVr~!r sin(!rt)� == jm 1Xn=�1 Jjnj 2eVr~!r ! sin(!J + n!r)t: (100)Jjnj is the Bessel fun tion of integer index.To interpret the experiments one must take into a ount that a non-zero voltage a ross the jun tion auses also a dissipative normal urrent In = V=R; where R is the resistan e of the jun tion for normalele trons.The experimental issue depends on the oupling in of the radio frequen y.1 If the impedan e ofthe radio sour e is small ompared to that of the jun tion, we have a voltage-sour e situation, andthe total urrent through the jun tion averaged over radio frequen ies isI = Is + V=R; Is = jsF: (101)1S. Shapiro, Phys. Rev. Lett. 11, 80 (1963). 35

I V=RShapiro spikes for!J + n!r = 0

d. . Josephson spike, Im ~!r=2eV

Figure 31: Josephson urrent vs. voltage in the voltage-sour e situation.If the impedan e of the radio sour e is large ompared to the impedan e of the Josephson jun tion,as is usually the ase, we have a urrent-sour e situation, where the fed-in total urrent determinesthe voltage a ross the jun tion:V = ~2e d dt = R(I � Is) = R(I � Im sin ): (102)�I �V =RIm

�V

Shapiro steps

Figure 32: Josephson urrent vs. voltage in the urrent-sour e situation.The a. . Josephson e�e t yields a possibility of pre ise measurements of h=e.36

7 MICROSCOPIC THEORY: THE FOCK SPACEThe S hr�odinger wavefun tion of an (isolated) ele tron is a fun tion of its position, r; and of thedis rete spin variable, s : � = �(r; s): Sin e there are only two independent spin states for an ele tron| the spin omponent with respe t to any (single) hosen axis may be either up (") or down (#) |,s takes on only two values, + and �, hen e � may be thought as onsisting of two fun tions�(r; s) = � �(r;+)�(r;�) � (103)forming a spinor fun tion of r: The expe tation value of any (usually lo al) one-parti le operatorA(r; s; r0; s0) = Æ(r � r0)A(r; s; s0) ishAi =Xs;s0 Z d3r��(r; s)A(r; s; s0)�(r; s0): (104)We will often use a short-hand notation x � (r; s).The S hr�odinger wavefun tion (x1 : : : xN ) of a (fermioni ) many-parti le quantum state must betotally antisymmetri with respe t to parti le ex hange (Pauli prin iple):(: : : xi : : : xk : : :) = �(: : : xk : : : xi : : :): (105)For a pie e of a solid, N � 1023, this fun tion is totally in omprehensible and pra ti ally ina essible:For any set of N values, + or � for ea h si; it is a fun tion of 3N positional oordinates. Althoughby far not all of those 2N fun tions are independent | with the symmetry property (105) the +- and�-values an always be brought to an order that all �-values pre ede all +-values, hen e we have onlyto distinguish 0, 1, 2, . . . , N �-values, that is (N + 1) ases | and although ea h of those (N + 1)fun tions need only be given on a ertain se tor of the 3N -dimensional position spa e | again withthe symmetry property (105) ea h fun tion need only be given for a ertain order of the parti le oordinates for all si = � and for all si = + parti les |, it is lear that even if we would be ontentwith 10 grid points along ea h oordinate axis we would need � 101023 grid points for a very rudenumeri al representation of that -fun tion. Nevertheless, for formal manipulations, we an introdu ein a systemati manner a fun tional basis in the fun tional spa e of those horrible -fun tions.7.1 Slater determinantsConsider some (for the moment arbitrarily hosen) omplete orthonormal set of one-parti le spinorfun tions , �l(x); (�lj�l0 ) =Xs Z d3r��l (r; s)�l0(r; s) = Æll0 ;Xl �l(x)��l (x0) = Æ(x � x0) = Æss0Æ(r � r0): (106)They are ommonly alled (spinor-)orbitals. The quantum number l refers to both the spatial andthe spin state and is usually already a multi-index (for instan e (nlm�) for an atomi orbital or (k�)for a plane wave), and we agree upon a ertain on e and forever given linear order of those l-indi es.Choose N of those orbitals, �l1 ; �l2 ; : : : ; �lN ; in as ending order of the li and form the determinant�L(x1 : : : xN ) = 1pN ! det k�li(xk)k: (107)L = (l1 : : : lN ) is a new (hyper-)multi-index whi h labels an orbital on�guration. This determinant ofa matrix aik = �li(xk) for every point (x1 : : : xN ) in the spin-position spa e has the proper symmetryproperty (105). In view of (106) it is normalized, if all li are di�erent, and it would be identi ally37

zero, if at least two of the li would be equal (determinant with two equal raws): Two fermions annotbe in the same spinor-orbital. This is the ompared to (105) very spe ial ase of the Pauli prin iple(whi h is the ommonly known ase).Now, given a omplete set of orbitals (106), we mention without proof that all possible orbital on�gurations of N orbitals (107) form a omplete set of N -fermion wavefun tions (105), that is, anywavefun tion (105) may be represented as(x1 : : : xN ) =XL CL�L(x1 : : : xN ) (108)with ertain oeÆ ients CL; PL jCLj2 = 1 (` on�guration intera tion').For any operator whi h an be expanded into its one-parti le, two-parti le- and so on parts as forinstan e a Hamiltonian with (possibly spin-dependent) pair intera tions,H =Xi hsis0i(ri) + 12Xi6=j wsis0i;sjs0j (ri; rj); (109)the matrix with Slater determinants isHLL0 = h�LjHj�L0i = Xi XP (lPijhjl0i) (�1)jPj Yj(6=i) ÆlPj l0j ++12Xi6=jXP (lPilPj jwjl0j l0i) (�1)jPj Yk(6=i;j) ÆlPkl0k ; (110)where P is any permutation of the subs ripts i; j; k; and jPj is its order. The matrix elements arede�ned as (lijhjl0i) =Xss0 Z d3r��li (r; s)hss0(r)�l0i(r; s0); (111)and(lilj jwjl0j l0i) = Xsis0i;sjs0j Z d3rid3rj ��li(ri; si)��lj (rj ; sj) wsis0i;sjs0j (ri; rj) �l0j (rj ; s0j)�l0i (ri; s0i): (112)(Note our onvention on the order of indi es whi h may di�er from that in other textbooks but leadsto a ertain anoni al way of writing of formulas later on.) The �rst line on the r.h.s. of (110) isnon-zero only if the two on�gurations L and L0 di�er at most in one orbital, and the sum over allpermutations P has only one non-zero term in this ase, determining the sign fa tor for that matrixelement. The se ond line is non-zero only if the two on�gurations di�er at most in two orbitals, andthe sum over all permutations has two non-zero terms in that ase: if P is a perturbation with lPk = l0kfor all k 6= i; j, then the orresponding ontribution is (1=2)[(lPilPj jwjl0j l0i) � ((lPj lPijwjl0j l0i)℄(�1)jPj.For L = L0 and P = identity (and sometimes also in the general ase) the �rst matrix element is alled dire t intera tion and the se ond one ex hange intera tion.7.2 The Fo k spa eUp to here we onsidered representations of quantum me hani s by wavefun tions with the parti lenumber N of the system �xed. If this number is ma ros opi ally large, it annot be �xed at a singlede�nite value in experiment. Zero mass bosons as e.g. photons may be emitted or absorbed in systemsof any s ale. (In a relativisti des ription any parti le may be reated or annihilated, possibly togetherwith its antiparti le, in a va uum region just by applying energy.) From a mere te hni al point ofview, quantum statisti s of identi al parti les is mu h simpler to formulate with the grand anoni alensemble with varying parti le number, than with the anoni al one. Hen e there are many goodreasons to onsider quantum dynami s with hanges in parti le number.38

In order to do so, we start with building the Hilbert spa e of quantum states of this wider frame:the Fo k spa e. The onsidered up to now Hilbert spa e of all N -parti le states having the appro-priate symmetry with respe t to parti le ex hange will be denoted by HN . In the last subse tion weintrodu ed a basis f�Lg in HN . Instead of spe ifying the multi-index L as a raw of N indi es li wemay denote a basis state by spe ifying the o upation numbers ni (being either 0 or 1) of all orbitalsi: jn1 : : : ni : : :i; Xi ni = N: (113)Our previous determinantal state (107) is now represented asj�Li = j0 : : : 01l10 : : : 01l20 : : : 01lN0 : : :i:Two states (113) not oin iding in all o upation numbers ni are orthogonal. HN is the ompletelinear spa e spanned by the basis ve tors (113), i.e. the states of HN are either linear ombinationsP j�LiCL of states (113) (with the sum of the squared absolute values of the oeÆ ients CL equal tounity) or limits of Cau hy sequen es of su h linear ombinations. (A Cau hy sequen e is a sequen efjnig with limm;n!1 hm � njm � ni = 0. The in lusion of all limits of su h sequen esinto HN means realizing the topologi al ompleteness property of the Hilbert spa e, being extremelyimportant in all onsiderations of limits. This ompleteness of the spa e is not to be onfused withthe ompleteness of a basis set f�ig.The extended Hilbert spa e F (Fo k spa e) of all states with the parti le number N not �xed isnow de�ned as the ompleted dire t sum of all HN . It is spanned by all state ve tors (113) for all Nwith the above given de�nition of orthogonality retained, and is ompleted by orresponding Cau hysequen es, just as the real line is obtained from the rational line by ompleting it with the help ofCau hy sequen es of rational numbers.Note that F now ontains not only quantum states whi h are linear ombinations with varyingni so that ni does not have a de�nite value in the quantum state (o upation number u tuations),but also linear ombinations with varying N so that now quantum u tuations of the total parti lenumber are allowed too. (For bosoni �elds as e.g. laser light those quantum u tuations an be omeimportant experimentally even for ma ros opi N .)7.3 O upation number representationWe now ompletely abandon the awful wavefun tions (105) and will ex lusively work with the o - upation number eigenstates (113) and matrix elements between them. The simplest operators arethose whi h provide just a transition between basis states (113) whi h are as lose to ea h other aspossible: those whi h di�er in one o upation number only.The de�nition of these reation and annihilation operators for fermions must have regard to the an-tisymmetry of the quantum states and to Pauli's ex lusion prin iple following from this antisymmetry.They are de�ned as ij : : : ni : : :i = j : : : ni � 1 : : :ini (�1)Pj<i nj ; (114) yi j : : : ni : : :i = j : : : ni + 1 : : :i (1� ni) (�1)Pj<i nj : (115)The usefulness of the sign fa tors will be ome lear below. By onsidering the matrix elements withall possible o upation number eigenstates (113), it is easily seen that these operators have all theneeded properties, do parti ularly not reate non-fermioni states (that is, states with o upationnumbers ni di�erent from 0 or 1 do not appear: appli ation of i to a state with ni = 0 gives zero,and appli ation of yi to a state with ni = 1 gives zero as well). The i and yi are mutually Hermitian onjugate, obey the key relationsnij : : : ni : : :i � yi ij : : : ni : : :i = j : : : ni : : :ini (116)and [ i; yj ℄+ = Æij ; [ i; j ℄+ = 0 = [ yi ; yj ℄+ (117)39

with the anti ommutator [ i; yj ℄+ = i yj + yj i de�ned in standard way. Reversely, the anoni alanti ommutation relations (117) de�ne all the algebrai properties of the -operators and moreoverde�ne up to unitary equivalen e the Fo k-spa e representation (114, 115). (There are, however, vast lasses of further representations of those algebrai relations with a di�erent stru ture and not unitaryequivalent to the Fo k-spa e representation.)The basis (113) of the Fo k spa e is systemati ally generated out of a single basis ve tor, theva uum state ji � j0 : : : 0i (with N=0) by applying y-operators:jn1 : : : ni : : :i = : : : yi : : : y1ji: (118)Observe again the order of operators de�ning a sign fa tor in view of (117) in agreement with thesign fa tors of (114, 115). Sin e produ ts of a given set of N y-operators written in any order agreewith ea h other up to possibly a sign, all possible expressions (118) do not generate more di�erentbasis ve tors than those of (107) with the onvention on the order of the li as agreed upon there.Hen eforth, by using (118) we need not bother any more about the given linear order of the orbitalindi es.With the help of the -operators, any linear operator in the Fo k spa e may be expressed. It isnot diÆ ult to demonstrate that the HamiltonianH =Xij yi (ijhjj) j + 12Xijkl yi yj(ijjwjkl) k l (119)has the same matrix elements with o upation number eigenstates (113) as the Hamiltonian (109))has with determinantal states in (110). Be ause of the one-to-one orresponden e between the deter-minantal states (107) and the o upation number eigenstates and be ause both span the Fo k spa e,by linearity the Hamiltonians (109) and (119) are equivalent. The building prin iple of the equivalentof any linear operator given in the S hr�odinger representation is evident from (119).The S hr�odinger wavefun tion of a bosoni many-parti le quantum state must be totally symmetri with respe t to parti le ex hange (omission of the minus sign in (105)). The determinants are thento be repla ed by symmetrized produ ts (permanents), with a slightly more involved normalizationfa tor. The orbitals may now be o upied with arbitrary many parti les: ni = 0; 1; 2; : : : : This asemay be realized with bosoni reation and annihilation operatorsbij : : : ni : : :i = j : : : ni � 1 : : :ipni; (120)byi j : : : ni : : :i = j : : : ni + 1 : : :ipni + 1; (121)nij : : : ni : : :i � byi bij : : : ni : : :i = j : : : ni : : :ini: (122)with the anoni al ommutation relations[bi; byj ℄� = Æij ; [bi; bj ℄� = 0 = [byi ; byj ℄�: (123)The basis states of the Fo k spa e are reated out of the va uum a ording tojn1 : : : ni : : :i = �by1�n1pn1! � � � �byi�nipni! � � � ji: (124)The order of these operators in the produ t does not make any di�eren e. The hoi e of fa torson the r.h.s. of (120, 121) not only ensures that (122) holds but also ensure the mutual Hermitian onjugation of bi and byi .7.4 Field operatorsA spatial representation may be introdu ed in the Fo k spa e by de�ning �eld operators (x) =Xi �i(x)ai; y(x) =Xi ��i (x)ayi ; (125)40

where the ai mean either fermioni operators i or bosoni operators bi. The �eld operators (x) and y(x) obey the anoni al (anti-) ommutation relations[ (x); y(x0)℄� = Æ(x� x0); [ (x); (x0)℄� = 0 = [ y(x); y(x0)℄�: (126)They provide a spatial parti le density operatorn(r) =Xs y(r; s) (r; s) (127)having the propertieshn(r)i =Xij Xs ��i (r; s)hayi aji�j(r; s); Z d3r n(r) =Xi ayi ai: (128)These relations are readily obtained from those of the reation and annihilation operators, and bytaking into a ount the ompleteness and orthonormality (106) of the orbitals �i.In terms of �eld operators, the Hamiltonian (109) or (119) readsH = Xss0 Z d3r y(r; s) hss0(r) (r; s0) ++12 Xs1s01s2s02 Z d3r1d3r2 y(r1; s1) y(r2; s2) ws1s01;s2s02(r1; r2) (r2; s02) (r1; s01): (129)It is obtained by ombining (119) with (125) and (111, 112).

41

8 MICROSCOPIC THEORY: THE BCS MODELThe great advantage of the use of reation and annihilation or �eld operators lies in the fa t that we an use them to manipulate quantum states in a physi ally omprehensible way without expli itlyknowing the wavefun tion. We even an think of modi�ed operators of whi h we know little more thantheir algebrai properties. The point is that the o upation number formalism applies for every orbitalset (106). The transition from one set of operators obeying anoni al (anti-) ommutation relations toanother su h set is alled a anoni al transformation in quantum theory.8.1 The normal Fermi liquid as a quasi-parti le gasA normal ondu ting Fermi liquid has a fermioni quasi-parti le ex itation spe trum whi h behavesvery mu h like a gas of independent parti les with energies �k (for the sake of simpli ity we assume itisotropi in k-spa e although this assumption is not essential here). Non-intera ting fermions wouldhave a ground state with all orbitals with � < � o upied and all orbitals with � > � empty; � isthe hemi al potential. By adding or removing a fermion with � = � new ground states with N � 1fermions are obtained. By adding a fermion with � > � an ex ited state is obtained with ex itationenergy � � �. By removing a fermion with � < � | that is, reating a hole in the original groundstate | an ex ited state is obtained with ex itation energy j� � �j: �rst lift the fermion to the level� and then remove it without hanging the hara ter of the state any more (Figs. 33 and 34).0

�� �kkF

�kquasi-ele tronquasi-holeFIG. 33: Creation of an ex ited ele tron and of a hole, resp.

�kkF

j�k � �jFIG. 34: Ex itation spe trum of a Fermi gas.The ground state j0i of a normal metal has mu h the same properties: ondu tion ele trons with� > � and holes with � < � may be ex ited with ex itation energies as above. These are not theoriginal ele trons making up the metal together with the atomi nu lei. Rather they are ele trons ormissing ele trons surrounded by polarization louds of other ele trons and nu lei in whi h nearly allthe Coulomb intera tion is absorbed. We do not pre isely know these ex itations nor do we know theground state j0i (although a quite elaborate theory exists for them whi h we ignore here). We justassume that they may be represented by fermioni operators with properties like those in the gas:�k < � : yk� j0i = 0; yk� k� j0i = j0i; (130)�k > � : k� j0i = 0; k� yk� j0i = j0i; (131)42

k is the waveve tor and � the spin state of the quasi-parti le.Sin e all intera tions present in the ground state j0i are already absorbed in the quasi-parti leenergies �k, only ex ited ondu tion ele trons or holes exert a remainder intera tion. Hen e, we maywrite down an e�e tive Hamiltonian~H = Xk� yk�(�k � �) k� + 12 �k>�; �k0>�Xk�;k0�0; q yk+q� yk0�q�0 wkk0q k0�0 k� ++ �k>�; �k0<�Xk�;k0�0; q yk+q� k0�q�0 wkk0q yk0�0 k� ++ 12 �k<�; �k0<�Xk�;k0�0; q k+q� k0�q�0 wkk0q yk0�0 yk� : (132)The matrix elements in the three lines are qualitatively di�erent: they are predominantly repulsivein the �rst and last line and attra tive in the se ond line; ele trons and holes have opposite harges.With the relations (130, 131) one �nds easilyh0j ~H j0i = �k<�Xk� (�k � �) = ~E0 ~H j0i = j0i ~E0: (133)We subtra t this onstant energy from the Hamiltonian and haveH � ~H � ~E0; h0jHj0i = 0 H j0i = 0: (134)We may reate a quasi-parti le above � in this ground state:�k1 > � : jk1�1i = yk1�1 j0i; H jk1�1i = jk1�1i (�k1 � �): (135)The last relation is easily veri�ed with our previous formulas. For k� 6= k1�1, we have yk� k� yk1�1 = yk1�1 yk� k� , and together with k1�1 yk1�1 j0i = j0i one �nds the above result. Likewise�k1 < � : jk1�1i = k1�1 j0i; H jk1�1i = jk1�1i j�k1 � �j (136)is obtained. Here, yk1�1 k1�1 k1�1 = 0, so that one term has to be removed from the sum of (133).Hen e, the single-parti le ex itation spe trum of our e�e tive Hamiltonian above the state j0i is just�k = j�k � �j; (137)and the ex ited states are jk1�1i of (135, 136), whatever the wavefun tion of j0i might be.To say the truth, this all is only approximately right. There are no fermioni operators for whi hthe relations (130, 131) hold true exa tly for the true ground state j0i. Therefore, the �rst relation(133) and the last relations (135, 136) are also not rigorous. The quasi-parti les have a �nite lifetimewhi h may be expressed by omplex energies �k. However, for j�k j � � the approximation is quitegood in normal, weakly orrelated metals.Consider now a state with two ex ited parti les:jk1�1 k2�2i = yk1�1 yk2�2 j0i: (138)To be spe i� we onsider two ex ited ele trons, the ases with holes or with an ele tron and a holeare ompletely analogous. The appli ation of the e�e tive Hamiltonian yieldsH yk1�1 yk2�2 j0i = yk1�1 yk2�2 j0i (�k1 + �k2) +Xq yk1+q�1 yk2�q�2 j0i wk1k2q: (139)43

The intera tion term is obtained with the rule ki�i yki�i j0i = j0i. One ontribution appears fromk� = k1�1; k0�0 = k2�2; and another ontribution � yk2+q�2 yk1�q�1 j0i wk2k1q from k0�0 = k1�1;k� = k2�2: The minus sign in this ontribution is removed by anti ommuting the two y-operators,and then, by repla ing q with �q under the q-sum and observing wk2k1�q = wk1k2q whi h derivesfrom w(r1; r2) = w(r2; r1), this se ond ontribution is equal to the �rst one, when e omitting thefa tor (1/2) in front. (This is how ex hange terms appear automati ally with -operators sin e theiranti ommutation rules automati ally retain the antisymmetry of states.) For the simpli ity of writingwe omitted here and in (132) the spin dependen e of the intera tion matrix element. It is alwayspresent in the e�e tive quasi-parti le intera tion, and it may always be added afterwards without onfusion. (We might introdu e a short-hand notation k for k�.) The e�e tive intera tion of twoquasi-parti les with equal spin di�ers from that of two quasi-parti les with opposite spin. The spin- ip s attering of quasi-parti les | an intera tion with hanging �1 and �2 into �01 and �02 | mayoften be negle ted. Then, the q-sum of (139) need not be ompleted by additional spin sums.8.2 The Cooper problem1From (139) it an be seen that the state (138) is not any more an eigenstate of the e�e tive Hamiltonian,not even within the approximations made in the previous subse tion. The two ex ited quasi-parti lesintera t and thus form a orrelated pair state. We try to �nd this pair state of lowest energy fortwo ele trons (� > 0) within our approximate approa h. Sin e we expe t that this state is formedout of quasi-parti le ex itations with energies � � 0 our approximations annot be riti al. (Thequasi-parti le lifetime be omes in�nite for j�j ! 0:) We expe t that the state lowest in energy haszero total momentum, hen e we build it out of quasi-parti le pairs with k2 = �k1:j i =Xk ak jk� � k�0i: (140)where we assume a �xed ombination of � and �0 and the yet unknown expansion oeÆ ients todepend on k only, be ause the sought state is to be expe ted to have a de�nite total spin. (Re allthat we are onsidering an isotropi metal in this hapter.) We want that this pair state j i is aneigenstate of H :H j i = j iE; H j i =Xk yk� y�k�0 j0i 2�kak +Xkq yk+q� y�k�q�0 j0i wk�kq ak: (141)Multiply the last relation with h0j �k0�0 k0� and observe h0j �k0�0 k0� yk� y�k�0 j0i = h0j �k0�0(Ækk0 � yk� k0�) y�k�0 j0i = Ækk0h0j(Ækk0 � y�k�0 �k0�0)j0i � h0j �k0�0 yk�(Æ�kk0Æ��0 � y�k�0 k0�)j0i = Ækk0 �h0j(Æ�kk0Æ��0 � yk� �k0�0)Æ�kk0Æ��0 j0i = Ækk0 � Æ�kk0Æ��0 to obtainE�ak0 � a�k0Æ��0� = 2�k0�ak0 � a�k0Æ��0�+Xq wk0�q;�k0+q;q�ak0�q � a�k0+qÆ��0�: (142)In the last term, we also used again w�k0�q;k0+q;q = wk0+q;�k0�q;�q and then repla ed the sum overq by a sum over �q.Due to the isotropy of our problem we expe t the solution to be an angular momentum eigenstate,hen e ak should have a de�nite parity. It is immediately seen that a non-trivial solution with evenparity a�k = ak (even angular momentum) is only possible, if Æ��0 = 0, that is for a singlet �0 = ��.For a spin triplet �0 = � only a non-trivial solution with odd parity (odd angular momentum) ispossible. To be spe i� , onsider the singlet ase. (The triplet ase is analogous.) Assumeak = akYlm(k=k) (143)1L. N. Cooper, Phys. Rev. 104, 1189 (1956). 44

with even l: In (142), rename k0 ! k, k0 � q ! k0. The matrix element wk0;�k0;k�k0 determines thes attering amplitude from states k;�k into states k0;�k0. we use an expansionwk0;�k0;k�k0 =Xlm �lwlkwl�k0Ylm(k)Y �lm(k0): (144)This redu es (142) to ak = �lwlkCElm � 2�k ; C =Xk0 wl�k0ak0 : (145)Inserting the left relation into the right one yields1 = �l Xk jwlk j2 1Elm � 2�k = �lF (Elm): (146)The sought lowest pair energy orresponds to the lowest solution of F (Elm) = 1=�l.

0 2�k EF

Elm1�l < 01�l > 0

Figure 35: The fun tion F (E) from Eq. (146).

The fun tion F (Elm)has poles for Elm = 2�kwhere it jumps from �1to +1. Re all that the�k-values are all positiveand start from zero. ForElm ! �1, F (Elm) ap-proa hes zero from nega-tive values. Hen e, if �l >0, then the lowest solu-tion Elm of (146) is posi-tive and the ground statej0i of the normal metalis stable. If at least one�l-value is negative (at-tra tive intera tion), thenthere is unavoidably a neg-ative solution Elm of (146):the `ex ited pair' has nega-tive energy and the normalground state j0i is unsta-ble against forming of pairsof bound quasi-parti les,no matter how small j�ljis (how weak the attra -tive intera tion is). Pairsare spontaneously formedand the ground state re on-stru ts. This is the ontentof Cooper's theorem.If the intera tion is ut of at some energy ! ,wlk = � 1 for 0 < �k < ! 0 elsewhere ; (147)and the density of states for �k is nearly onstant in this interval, N(�) = N(0); then, with negativeElm, 45

Xk jwlk j2 1Elm � 2�k = �N(0) Z ! 0 d� 1jElmj+ 2� = �N(0)2 ln" jElm + 2! jElmj #;hen e, jElmj = 2! exph 2N(0)j�lji� 1 : (148)This yields jElmj � 8><>: 2! exp"� 2N(0)j�lj#N(0)j�lj! for N(0)j�lj �� 1 (149)in the weak and strong oupling limits. In this hapter we only onsider the weak oupling limit wherejElmj is exponentially small.The whole analysis may be repeated for the ase where the pair has a non-zero total momentumq. In that ase the denominator of (146) is to be repla ed with Elm(q)� �k+q=2� ��k+q=2 where nowjk� q=2j must be larger than kF . For small q; this ondition redu es the density of states in e�e t inan interval of thi kness j��=�kjq=2 = vF q=2 at the lower �-integration limit; vF is the Fermi velo ity.The result is Elm(q) � Elm + vF q=2: (150)In the weak oupling limit, Elm(q) an only be negative for exponentially small q.We performed the analysis with a pair of parti les. It an likewise be done with a pair of holeswith an analogous result.8.3 The BCS HamiltonianFr�ohli h1 was the �rst to point out that the ele tron-phonon intera tion is apable of providing ane�e tive attra tion between ondu tion ele trons in the energy range of phonon energies.From Cooper's analysis it follows that, if there is a weak attra tion, it an only be e�e tive forpairs with zero total momentum, that is, between k and �k. With the assumption that the attra tionis in the l = 0 spin singlet hannel, this led Bardeen, Cooper and S hrie�er2 to the simple modelHamiltonian HBCS =Xk� yk�(�k � �) k� � gV ��! <�k;�k0<�+! Xkk0 yk0" y�k0# �k# k": (151)Here, g > 0 is the BCS oupling onstant, and V is the normalization volume. Sin e the density ofplane-wave states in k-spa e is V=(2�)3: Pk = V=(2�)3 R d3k, the matrix element of an n-parti leintera tion (appearing in an n-fold k-sum) must be proportional to V �(n�1) in order that the Hamil-tonian is extensive (� V ). The modeled attra tive intera tion is assumed in an energy range of width2! around the hemi al potential (Fermi level in the ase T = 0), where ! is a hara teristi phononenergy for whi h the Debye energy of the latti e an be taken.The state j0i of (130, 131) annot any more be the ground state of this Hamiltonian sin e Cooper'stheorem tells us that this state is unstable against spontaneous formation of bound pairs with the gainof their binding energy. The problem to solve is now to �nd the ground state and the quasi-parti lespe trum of the BCS-Hamiltonian. This problem was solved by Bardeen, Cooper and S hrie�er, and,shortly thereafter and independently by means of a anoni al transformation, by Bogoliubov andValatin. Bardeen, Cooper and S hrie�er thus provided the �rst mi ros opi theory of super ondu -tivity, 46 years after the dis overy of the phenomenon.1H. Fr�ohli h, Phys. Rev. 79, 845 (1950).2J. Bardeen, L. N. Cooper, and J. R. S hrie�er, Phys. Rev. 108, 1175 (1957).46

8.4 The Bogoliubov-Valatin transformation1Suppose the ground state ontains a bound pair. Ex iting one parti le of that pair leaves its partnerbehind, and hen e also in an ex ited state. If one wants to ex ite only one parti le, one must annihilatesimultaneously its partner. Led by this onsideration, for the quasi-parti le operators in the groundstate of (151) an ansatz bk" = uk k" � vk y�k#; bk# = uk k# + vk y�k"is made. uk and vk are variational parameters. Again we onsider the isotropi problem and hen etheir dependen e on k = jkj only. The reason of the di�erent signs in the two relations be omes lear in a minute. Sin e for ea h orbital annihilated by k"; �k#; bk"; b�k# an r-independent phasefa tor may be arbitrarily hosen, uk and vk may be assumed real without loss of generality. TheseBogoliubov-Valatin transformations together with their Hermitian onjugate may be summarized asbk� = uk k� � �vk y�k��; byk� = uk yk� � �vk �k�� : (152)We want these transformations to be anoni al, that is, we want the new operators bk� ; byk� again tobe fermioni operators. One easily al ulates[bk� ; bk0�0 ℄+ = �ukvk0��0[ k� ; y�k0��0 ℄+ + �[ y�k�� ; k0�0 ℄+� == �ukvk0��0Æk�k0Æ���0 + �Æ�kk0Æ���0� = �ukvkÆ�kk0 (�� + �) = 0:In the �rst equality it was already onsidered that annihilation and reation operators and y,respe tively, anti ommute among themselves. The analogous result for the by-operators is obtained inthe same way. The sign fa tor � in the transformation ensures that the anti ommutation is retainedfor the b- and by-operators, respe tively. Analogously,[bk� ; byk0�0 ℄+ = u2k[ k� ; yk0�0 ℄+ + ��0v2k[ y�k�� ; �k0��0 ℄+ = �u2k + v2k�Ækk0Æ��0 ;hen e the ondition u2k + v2k = 1 (153)ensures that the transformation is anoni al and the new operators are again fermioni operators.Multiplying the �rst relation (152) by uk, repla ing in the se ond one k� with �k��, multiplyingit with �vk , and then adding both results yields with (153) the inverse transformation k� = ukbk� + �vk by�k��; yk� = ukbyk� + �vk b�k�� : (154)Observe the reversed sign fa tor.The next step is to transform the Hamiltonian (151). With yk� k� = �uk byk� + �vk b�k����ukbk� + �vk by�k��� == u2kbyk� bk� + v2k b�k�� by�k�� + �ukvk�byk� by�k�� + b�k�� bk��and the anti ommutation rules it is easily seen that the single-parti le part of the BCS-Hamiltoniantransforms into2Xk (�k � �)v2k +Xk (�k � �)(u2k � v2k)X� byk� bk� + 2Xk (�k � �)ukvk�byk"by�k# + b�k#bk"�:It has also been used that under the k-sum k may be repla ed by �k: Further, withBk = �k# k" = �ukb�k# � vk byk"��uk bk" + vk by�k#� == u2kb�k#bk" � v2k byk"by�k# + ukvk�b�k#by�k# � byk"bk"�; (155)1N. N. Bogoliubov, Nuovo Cimento 7, 794 (1958); J. G. Valatin, Nuovo Cimento 7, 843 (1958).47

the full transformed BCS-Hamiltonian readsHBCS = 2Xk (�k � �)v2k +Xk (�k � �)(u2k � v2k)X� byk� bk� ++2Xk (�k � �)ukvk�byk"by�k# + b�k#bk"�� gV Xkk0 Byk0Bk; (156)where for brevity we omitted the bounds of the last sum.Now, we introdu e the o upation number operatorsnk� = byk� bk� (157)of the b-operators and assume analogous to (130, 131) that for properly hosen b-operators the groundstate j0i of the BCS-Hamiltonian is an o upation number eigenstate. Then, its energy is found tobe E = 2Xk (�k � �)v2k +Xk (�k � �)(u2k � v2k)(nk" + nk#)� gV "Xk ukvk(1� nk" � nk#)#2: (158)The nk� are the eigenvalues (0 or 1) of the o upation number operators (157) in the ground statej0i.This energy expression still ontains the variational parameters uk and vk whi h are onne ted by(153), when e �vk=�uk = �uk=vk. For given o upation numbers, (158) has its minimum for�E�uk = "�4(�k � �)uk + 2�u2k � v2kvk #�1� nk" � nk#� = 0;� = gV Xk ukvk�1� nk" � nk#�: (159)Hen e, uk and vk are determined by (153) and2(�k � �)ukvk = ��u2k � v2k�: (160)Their ombination yields a biquadrati equation with the solutionu2kv2k � = 12"1� (�k � �)p(�k � �)2 +�2 #; 2ukvk = �p(�k � �)2 +�2 : (161)Insertion into (159) results in the self- onsisten y ondition1 = g2V Xk 1� nk" � nk#p(�k � �)2 +�2 ; (162)whi h determines � as a fun tion of the BCS oupling onstant g, the dispersion relation �k of thenormal state -quasi-parti les (in essen e the Fermi velo ity), and the o upation numbers nk� of theb-quasi-parti les of the super ondu ting state (in essen e the temperature as seen later).48

� �k12�vk uk

Figure 36: The fun tions u and v.

The parametersuk and vk are de-pi ted in the �gureon the right. For-mally, inter hanginguk with vk wouldalso be a solutionof the biquadrati equation. It is, how-ever, easily seen thatthis would not leadto a minimum of(158). With (161) itis readily seen thatthe se ond sum of(158) is positive def-inite. Hen e, the absolute minimum of energy (ground state) is attained, if all o upation numbers ofthe b-orbitals are zero.For the ground state, the self- onsisten y ondition (162) redu es to1 = g2V Xk 1p(�k � �)2 +�20 = gN(0)2 Z ! �! d! 1p!2 +�20 = gN(0)2 lnp!2 +�20 + ! p!2 +�20 � ! �� gN(0)2 ln 4!2 �20resulting in �0 = 2! exp�� 1gN(0)� (163)for the value of � in the ground state (at zero temperature).If one repla es the last term �(g=V )P Byk0Bk of the transformed BCS-Hamiltonian (156) by themean-�eld approximation��Pk�Bk+Byk� (re all that � was introdu ed as � = (g=V )h0jPk Bkj0i, f. (155, 159)), than it is readily seen that the relation (160) makes the anomalous terms (terms bybyor bb) of this Hamiltonian vanish: In mean-�eld approximation the BCS-Hamiltonian is diagonalizedby the Bogoliubov-Valatin transformation, resulting inHm-f = 2Xk (�k � �)v2k +Xk� (�k � �)(u2k � v2k)byk� bk� � 2�Xk ukvk�1�X� byk� bk�� == onst. +Xk� h(�k � �)(u2k � v2k) + 2�ukvkibyk� bk� == onst. +Xk� �k� byk� bk� (164)with the b-quasi-parti le energy dispersion relation�k =p(�k � �)2 +�2 (165)obtained by inserting (161) into the se ond line of (164).49

kfF ��k � �

�kj�k � �j�Figure 37: Quasi-parti le dispersion relation in thesuper ondu ting state.

The dispersion relation �k together with thenormal state dispersion relation �k � � andthe normal state ex itation energy dispersionj�k � �j is depi ted on the right. It is seenthat the physi al meaning of � is the gap inthe b-quasiparti le ex itation spe trum of thesuper ondu ting state. The name bogolons isoften used for these quasi-parti les.The fa t that the Bogoliubov-Valatin trans-formation diagonalizes the BCS-Hamiltonianat least in mean-�eld approximation justi�esa posteriori our assumption that the groundstate may be found as an eigenstate of the b-o upation number operators. In the litera-ture, the key relations (160) are often derived asdiagonalizing the mean-�eld BCS-Hamiltonianinstead of minimizing the energy expression(158). In fa t both onne tions are equally im-portant and provide only together the solution of that Hamiltonian. Clearly, the BCS-theory basedon that solution is a mean-�eld theory.From (152) one ould arrive at the on lusion that a bogolon onsists partially of a normal-stateele tron and partially of a hole, and hen e would not arry an integer harge quantum. However, as�rst was pointed out by Josephson, the true b-quasi-parti le annihilation and reation operators are�k� = uk k� � �vkP y�k�� ; �yk� = uk yk� � �vk �k��P y; (166)where P annihilates and P y reates a bound pair with zero momentum and zero spin as onsideredin the Cooper problem. Its wavefun tion will be onsidered in the next hapter. Now, a bogolon isannihilated, that is, a hole-bogolon is reated, by partially reating a normal-state hole and partiallyannihilating an ele tron pair and repla ing it with a normal-state ele tron. The se ond part of thepro ess also reates a positive harge, when e the hole-bogolon (with jkj < kF ) arries an integerpositive harge quantum. Likewise, an ele tron-bogolon (with jkj > kF ) is reated partially by reatinga normal-state ele tron and partially by reating an ele tron pair and simultaneously annihilating anormal-state ele tron. Again, the bogolon arries an integer (negative) harge quantum. The P -operators make the b-bogolon be surrounded by a super ondu ting ba k ow of harge whi h ensuresthat an integer harge quantum travels with the bogolon. It is positive for jkj < kF and negative forjkj > kF .Of ourse, it remains to show that the new ground state j0i is indeed super ondu ting.

50

9 MICROSCOPIC THEORY: PAIR STATESIn mean-�eld BCS theory, the ground state is determined by the omplete absen e of quasi-parti les.With the properties of the Bogoliubov-Valatin transformation, this ground state is found to be the ondensate of Cooper pairs in plane-wave K = 0 states of their enters of gravity. By o upyingquasi-parti le states a ording to Fermi statisti s, thermodynami states of the BCS super ondu torare obtained.9.1 The BCS ground stateIn Se tion 8.D the BCS ground state j0i was assumed to be an o upation number eigenstate ofnk� = byk� bk� ; and the bk� were determined a ordingly. Then, it was found after (162) that allo upation numbers nk� are zero in the ground state. This implies that h0jbyk� bk� j0i = 0; andhen e bk� j0i = 0 (167)for all k�: The (mean-�eld) BCS ground state is the (uniquely de�ned) b-va uum. We show thatj0i =Yk �uk + vk yk" y�k#�ji (168)is the properly normalized state with properties (167). The normal metal ground state j0i of (130,131) is j0i = �k<�Yk� yk� ji = �k<�Yk yk" y�k#ji (169)and hen e has the form (168) too, with uk = 0 for �k < � and uk = 1 for �k > � and the oppositebehavior for vk.We now demonstrate the properties of (168). Sin ehj�uk + vk �k# k"��uk + vk yk" y�k#�ji = u2k + v2k = 1;j0i of (168) is properly normalized. Moreover,bk0"j0i = " Yk(6=k0)�uk + vk yk" y�k#�#�uk0 k0" � vk0 y�k0#��uk0 + vk0 yk0" y�k0#�ji == " Yk(6=k0)�uk + vk yk" y�k#�#uk0vk0� k0" yk0" y�k0# � y�k0#�ji == " Yk(6=k0)�uk + vk yk" y�k#�#uk0vk0��1� yk0" k0"� y�k0# � y�k0#�ji = 0 (170)and analogously bk0#j0i = 0. This ompletes our proof. Histori ally, Bardeen, Cooper and S hrie�ersolved the BCS model with the ansatz (168), before the work of Bogoliubov and Valatin.51

9.2 The pair fun tionNext we �nd the wavefun tions ontained in j0i: Re all, that yk� reates a ondu tion ele tron inthe plane-wave state � exp(ik � r)��(s) so that the �eld operator y(rs) is y(rs) =Xk� yk� exp(ik � r)��(s): (171)(That is, rs denotes the enter of gravity and the spin of the ele tron with its polarization loud whi htogether make up the ` ondu tion ele tron'.)The N -parti le wavefun tion ontained in j0i and depending on these variables is0(x1; : : : ; xN ) = hx1 : : : xN j0i = hj (xN ) � � � (x1) Yk �uk + vk yk" y�k#�ji: (172)For the sake of brevity we suppress the spinors �, and �nd0(x1; : : : ; xN ) == Xk1:::kNhj eikN �rN kN�N eikN�1�rN�1 kN�1�N�1 � � � eik1�r1 k1�1| {z }=0; if not ki�i 6=kj�j for i6=j Yk �1 + gk yk" y�k#�Yk uk ji == Xk1:::kN 0ei(k1�r1+���+kN �rN ) hj kN�N p� � � p k1�1 N=2Yk gkq yk"q y�k#| {z }sum over all possible ontra tions ji Yk uk! �� Xfk2ig0 eik2�(r1�r2)gk2�1��2 � � � eikN �(rN�1�rN )gkN �N�1��N � � � �! =� A�(r1 � r2)�singlet�(r3 � r4)�singlet � � ��(rN�1 � rN )�singlet (173)with �(�) �Xk gkeik��; gk = vkuk : (174)In the se ond line of (173), (171) was inserted for the (xi) of (172), and the uks were fa tored out ofthe produ t of (172) (leaving gk = vk=uk behind in the se ond item of the fa tors). The k-produ tsrun over all grid points of the (in�nite) k-mesh, e.g. determined by periodi boundary onditions forthe sample volume V , while the sum runs over all possible produ ts for sets of N disjun t k-values outof that mesh. This disjun t nature of the k-sums is indi ated by a dash at the sum in the followinglines. Expansion of the �rst k-produ t yields terms with 0, 2, 4, . . . y-operators of whi h only theterms with exa tly those N y-operators that orrespond to the N -operators of an item of the k-sumleft to the produ t produ e a non-zero result between the -va uum states hj � � � ji. These results aremost easily obtained by anti ommuting all annihilation operators to the right of all reation operatorsand are usually alled ontra tions; depending on the original order of the operators, ea h result is�1. The produ t over the uk, whi h multiplies ea h ontribution and whi h as previously runs overall in�nitely many k-values of the full mesh, yields a normalizing fa tor whi h is independent of thevalues of k1; : : : ;kN of the sums. Like ea h individual fa tor uk, it depends on the hemi al potential� and on the gap �0. At this point one must realize that the gk are essentially non-zero inside ofthe Fermi surfa e ( f. (161)). Hen e, the ontribution to 0(x1; : : : ; xN ) has a non-negligible valueonly for all ki-ve tors inside the Fermi surfa e, and this value in reases with an in reasing number ofsu h ki-ve tors and de reases again, if an appre iable number of ki-ve tors falls outside of the Fermisurfa e: the norm of (172) is maximal for N -values su h that the k2i o upy essentially all mesh pointsinside the Fermi surfa e. That is, this norm is non-negligible only for those N -values orresponding to52

the ele tron number in the original normal Fermi liquid: j0i is a grand- anoni al state with a sharpparti le-number maximum at the anoni al N -value. We did not tra e normalizing fa tors in (173)and use a sign of approximation in (174) again, assumig �(�) to be normalized.The result is a state j0i onsisting of pairs of ele trons in the geminal (pair-orbital) �(�)�singlet.In order to analyze what this pair-orbital �(�) looks like, re all that uk and vk may be written asfun tions of �0 and (�k � �)=�0 � ~vFk=�0 ( f. (161)). Hen e, gk � ~g(~vFk=�0), and�(�) � Z d3kgkeik�� � Z 10 dkk2gk Z 1�1 d�eik�� = Z 10 dkk2gk eik� � e�ik�ik� �� 1� Z 10 dkk~g�~vFk�0 � sin k� � 1� Z 10 dxx~g(x) sin��0x�~vF � � f(�=�0);�0 � 1�Æk = ~vF��0 : (175)By omparison with (173), � is the distan e ve tor of the two ele trons in the pair, their distan eon average being of the order of �0, while the pair-orbital � does not depend on the position R ofthe enter of gravity of the pair: with respe t to the enter of gravity the pair is delo alized, itis a plane wave with wave ve tor K = 0. Moreover, again due to (173), all N=2 � 1023 ele tronpairs o upy the same delo alized pair-orbital � in the BCS ground state j0i : this ma ros opi allyo upied delo alized (that is, onstant in R-spa e) pair-orbital is the ondensate wavefun tion of thesuper ondu ting state, and hen e the stru ture (173) of j0i ensures that the solution of the BCSmodel is a super ondu tor.For a real super ondu tor, the gap �0 an be measured (for instan e by measuring thermodynami quantities whi h depend on the ex itation spe trum or dire tly by tunneling spe tros opy. With theindependently determined Fermi velo ity vF of ele trons in the normal state, this measurement yieldsdire tly the average distan e �0 of the ele trons in a pair whi h an be ompared to the averagedistan e rs of to arbitrary ondu tion ele trons in the solid given by the ele tron density. For aweakly oupled type I super ondu tor this ratio is typi ally�0rs � 103 : : : 104: (176)In this ase, there are 109 : : : 1012 ele trons of other pairs in the volume between a given pair: Thereis a pair orrelation resulting in a ondensation of all ele trons into one and the same delo alized pairorbital in the super ondu ting state, however, the pi ture of ele trons grouped into individual pairswould be by far misleading.In order to reate a super urrent, the ondensate wavefun tion, that is, the pair orbital must beprovided with a phase fa tor �(�;R) � eiK�R (177)by repla ing the reation operators in (168) with yk+K=2" y�k+K=2#: Obviously, it must be K�0 � 1in order not to deform (and thus destroy) the pair orbital itself. Hen e, �0 has the meaning of the oheren e length of the super ondu tor at zero temperature.9.3 Non-zero temperatureThe transformed Hamiltonian (164) shows that the bogolons reated by operators �yk� of (166) andhaving an energy dispersion law �k of (165) are fermioni ex itations with harge e and spin � abovethe BCS ground state j0i: Sin e they may re ombine into Cooper pairs �, the hemi al potential ofthe Cooper pairs must be 2� where � is the hemi al potential of bogolons. Hen e, at temperature Tthe distribution of bogolons is nk" = nk# = 1e�k=kT + 1 : (178)53

The gap equation (162) then yields ( f. the analysis leading to (163))1 = g2V Xk 1� 2�e�k=kT + 1��1�k = gN(0)2 Z ! �! d!p!2 +�2 ep!2+�2=kT � 1ep!2+�2=kT + 1 == gN(0) Z ! 0 d!p!2 +�2 tanh p!2 +�22kT ! (179)where in the last equation the symmetry of the integrand with respe t to a sign hange of ! was used.For T ! 0, with the limes tanhx! 1 for x!1, (163) is reprodu ed.For in reasing temperature, the numerator of (162) de reases, and hen e � must also de rease. Itvanishes at the transition temperature T , when e1 = gN(0) Z ! 0 d!! tanh !2kT = gN(0) Z ! =2kT 0 dxx tanhx: (180)Integration of the last integral by parts yieldsZ ! =2kT 0 dxx tanhx = � Z ! =2kT 0 dx lnx osh2 x + ln ! 2kT tanh ! 2kT �� � Z 10 dx lnx osh2 x + ln ! 2kT = ln 4 � + ln ! 2kT = ln 2 ! �kT ;where ln = C � 0:577 is Euler's onstant. The se ond line is valid in the weak oupling asekT � ! and the �nal result for that ase iskT = 2 � ! exp(� 1gN(0)) = ��0 (181)

0��0

T TFigure 38: The gap as a fun tion of T .

with 2 =� � 1:13; and 2�0=kT =2�= � 3:52:The gap � as a fun tion oftemperature between zero and T must be al ulated numeri allyfrom (178). However, the simpleexpression�(T ) � �0s1�� TT �3 (182)is a very good approximation.From the energy expression(158), the thermodynami quanti-ties an be al ulated, on e �(T )and hen e �k(T ) is given. Themain results are the ondensationenergy at T = 0, B (0)22�0 = 12N(0)�20; (183)yielding the thermodynami riti al �eld at zero temperature, the spe i� heat jump at T ,Cs � CnCn � 1:43; (184)and the exponential behavior of the spe i� heat at low temperatures,Cs(T ) � T�3=2e��0=kT for T � T : (185)54

10 MICROSCOPIC THEORY: COHERENCE FACTORSAs already done in the last se tion, thermodynami states of a super ondu tor are obtained by o - upying quasi-parti le states with Fermi o upation numbers. External �elds, however, in most ases ouple to the -operator �elds. Due to the oupling of -ex itations in a super ondu tor interferen eterms appear in the response to su h �elds.10.1 The thermodynami stateLet fk�g be any given disjun t set of quasi-parti le quantum numbers. Then,jfk�gi = Yk�2fk�g �yk� j0i (186)is a state with those quasi-parti les ex ited above the super ondu ting ground state j0i: If there aremany quasi-parti le ex itations present, they intera t with ea h other and with the ondensate in theground state (the latter intera tion is in terms of the operators P yk� and k�P y), and this intera tionleads to the temperature dependen e of their energy dispersion law �k = p(�k � �)2 +�2 via thetemperature dependen e � = �(T ): The thermodynami state is�P = Xfk�g jfk�giQk�2fk�g fkZ hfk�gj; (187)where the summation is over all possible sets of quasi-parti le quantum numbers,fk = 1exp(�k=kT ) + 1 (188)is the Fermi o upation number, and Z = Z(T; �) is the partition fun tion determined bytr �P = Xfk�g Qk�2fk�g fkZ = 1: (189)As usual, the thermodynami expe tation value of any operator A is obtained as trA �P : For instan ethe average o upation number itself of a quasi-parti le in the state k0�0 isnk0�0 = tr��yk0�0 �k0�0 Xfk�g jfk�giQ fkZ hfk�gj� == Xfk�ghfk�gj�yk0�0 �k0�0 jfk�giQ fkZ = fk0tr �P = fk0 : (190)The state �k0�0 jfk�gi is obtained from the state jfk�gi by removing the quasi-parti le k0�0: Hen e,if one fa tors out fk0 , the remaining sum is again �P .10.2 The harge and spin moment densitiesThe operator of the q Fourier omponent of the harge density of ele trons in a solid isn(q) = �eXk� yk+q� k� = �eXk � yk+q" k" + y�k# �k�q#�: (191)That of the spin moment density (in z-dire tion) ism(q) = �BXk� yk+q�� k� = �BXk � yk+q" k" � y�k# �k�q#�: (192)55

The statisti al operator �P of a normal metalli state is omposed in analogy to (187) from eigenstatesjfk�gi of -operators. Then, in al ulating thermodynami averages, ea h item of the k�-sum of(191) and (192) is averaged independently. In the super ondu ting state, the items in parentheses ofthe last of those expressions are oupled and hen e they are not any more averaged independently:there appear ontributions due to their oherent interferen e in the super ondu ting states jfk�gi:Those ontributions appear in the response of the super ondu ting state to external �elds whi h oupleto harge and spin densities.Performing the Bogoliubov-Valatin transformation for the harge density operator yieldsXk � yk+q" k" + y�k# �k�q#� == Xk ��ujk+qjbyk+q" + vjk+qjb�k�q#��uk bk" + vk by�k#�++�ukby�k# � vk bk"��ujk+qjb�k�q# � vjk+qjbk+q"�� == Xk �ujk+qjukbyk+q"bk" + vjk+qjvk b�k�q#by�k# + ujk+qjvk byk+q"by�k# + vjk+qjukb�k�q#bk" ++ukujk+qjby�k#b�k�q# + vkvjk+qjbk"byk+q" � ukvjk+qjby�k#byk+q" � vkujk+qjbk"b�k�q#� == Xk ��ujk+qjuk � vjk+qjvk��byk+q"bk" + by�k#b�k�q#�++�ujk+qjvk + ukvjk+qj��byk+q"by�k# � bk"b�k�q#��In obtaining the last equality some operator pairs were anti ommuted whi h leads to the �nal resultn(q) = �eXk ��ujk+qjuk � vjk+qjvk��byk+q"bk" + by�k#b�k�q#�++�ujk+qjvk + ukvjk+qj��byk+q"by�k# � bk"b�k�q#��: (193)An analogous al ulation yieldsm(q) = �BXk ��ujk+qjuk + vjk+qjvk��byk+q"bk" � by�k#b�k�q#�++�ujk+qjvk � ukvjk+qj��byk+q"by�k# + bk"b�k�q#��: (194)The �rst line of these relations re e ts the above mentioned oupling between -states, and the se ondre e ts the oupling to the ondensate. Both lines ontain oheren e fa tors omposed of u and v.10.3 Ultrasoni attenuationAs an example of a �eld (external to the ele tron system) oupling to the harge density we onsiderthe ele tri �eld aused by a latti e phonon. The orresponding intera tion term of the Hamiltonianis HI = gpV Xq p!q�ay�q + aq�n(q); (195)where g is a oupling onstant relating the ele tri �eld of the phonon to its amplitude, V is thevolume, ! is the phonon frequen y and ay its reation operator.56

We onsider the attenuation of ultrasound with ~!q < �; then, in lowest order pair pro esses donot ontribute. A ording to Fermi's golden rule the phonon absorption rate may be written asRa(q) = 2�~ tr�HIÆ(Ef �Ei)HI �P � == 4�g2~V !qnqXk �ujk+qjuk � vjk+qjvk�2fk(1� fjk+qj)Æ(�jk+qj � �k � ~!q): (196)Here, Ef and Ei are the total energies of the states forming the HI -matrix elements and nq is thephonon o upation number of the thermodynami state whi h in this ase in extension of (187) also ontains phononi ex itations in thermi equilibrium. Half of the result of the last line is obtainedfrom the �rst term in the �rst line of (193). After renaming �k� q ! k0; the se ond term yields thesame result. The phonon emission rate is analogouslyRe(q) = 2�~ tr�HIÆ(Ef �Ei)HI �P � == 4�g2~V !qnqXk �ujk+qjuk � vjk+qjvk�2fjk+qj(1� fk)Æ(�jk+qj � �k � ~!q): (197)With fk(1� fjk+qj)� fjk+qj(1� fk) = fk � fjk+qj, the attenuation rate isdnqdt = �4�g2~V !qnqXk �ujk+qjuk � vjk+qjvk�2(fk � fjk+qj)Æ(�jk+qj � �k � ~!q): (198)The attenuation in the normal state isdnqdt = �4�g2~V !qnqXk (fk � fjk+qj)Æ(�jk+qj � �k � ~!q): (199)From a measurement of the di�eren e, �(T ) an be inferred.10.4 The spin sus eptibilityWith the intera tion Hamiltonian HI = �H(�q)m(q) + : :; (200)where H is an external magneti �eld, the expe tation value of the energy perturbation is obtainedas �E(q) = tr�HI(Ef �Ei)�1HI �P�: (201)The sus eptibility is�(q) = �d2�E(q)dH(�q) = �2�2BXk "�ujk+qjuk + vjk+qjvk�2 fjk+qj � fk�jk+qj � �k ++�ujk+qjvk � ukvjk+qj�2 1� fk � fjk+qj�k + �jk+qj #: (202)The sus eptibility drops down below T and vanishes exponentially for T ! 0.Coheren e fa tors appear in similar manners in many more response fun tions as in the nu learrelaxation time, in the diamagneti response, in the mi rowave absorption, and so on.57

Aknowledgement: I thank Dr. V. D. P. Servedio for preparing the �gures ele troni ally with great are.

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